EJERCICIOS DE INTEGRALES INDEFINIDAS · 1. ∫x2sen 3xdx = ∫ ∫ =− − − − − xcos 3xdx 3...
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EJERCICIOS DE INTEGRALES INDEFINIDAS
Transcript of EJERCICIOS DE INTEGRALES INDEFINIDAS · 1. ∫x2sen 3xdx = ∫ ∫ =− − − − − xcos 3xdx 3...
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
EJERCICIOS DE INTEGRALES INDEFINIDAS
1. ∫ dxx3senx2 = ∫ ∫−−=
−−− xdx3cosx3
2
3
x3cosxdx
3
x3cosx2
3
x3cosx 22
=
( )∗
( )∗ =
−−− ∫ dx3
x3sen
3
x3xsen
3
2
3
x3cosx2
=
++−− Kx3cos9
1
3
x3xsen
3
2
3
x3cosx2
= K27
x3cos2
9
x3xsen2
3
x3cosx2
++−−
2. ∫ +dx
9x
x2
= ∫ ++=+
K9xln2
1dx
9x
x2
2
1 22
3. ∫−
dxx1
x4
=
( ) ( )( )∫ ∫ +=
−=
−Kxarcsen
2
1dx
x1
x2
2
1dx
x1
x 2
2222
4. ( )∫ ++=+
= K1elndx1e
e)x(F x
x
x
si 2ln2K2K2ln2)0(F −=⇒=+⇒=
( ) 2ln21eln)x(F x −++=⇒
5. dx7x5
x23∫ +
−=
∫ ∫∫ =+
+−=+
+−=
++− dx
7x5
5
25
29x
5
2dx
7x5
1
5
29x
5
2dx
7x5
529
5
2
= K7x5ln25
29x
5
2 +++−
6. ( )∫ dx
x
xln 4
= ( ) Kxln5
1 5 +
7. ∫− dxex
2x3 = ∫ +−−=−−−−−
−−
K2
e
2
exdxxe2
2
1
2
ex22
2
2 xx2x
x2
3
x3cosv
xdx2du
xdx3sendv
xu 2
−=
==
=
3
x3senv
dxdu
xdx3cosdv
xu
=
==
=
2
2
x
x
2
e2
1v
xdx2du
dxxedv
xu
−
−
−=
==
=
8. ∫ dxex
1 x
1
2 = Ke x
1
+−
9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos
=
= Kxcos2xsenx2 ++
10. ( )∫ dxxlnsenx =
∫ ∫ =−=− dx)xcos(lnx2
1
2
)x(lnsenxdx
x
1)xcos(ln
2
x
2
)x(lnsenx 222
=
+− ∫ dx
x
1)x(lnsen
2
x
2
)xcos(lnx
2
1
2
)x(lnsenx 222
=
∫−− dx)x(lnxsen4
1
4
)xcos(lnx
2
)x(lnsenx 22
I4
1
4
)xcos(lnx
2
)x(lnsenxI
22
−−=
4
)xcos(lnx
2
)x(lnsenxI
4
5 22
−=
K4
)xcos(lnx
2
)x(lnsenx
5
4dx)x(lnxsen
22
+
−=∫
11. ∫ −+
dxxx
1x2
= ∫ ∫ +−+−=−
+−K1xln2xlndx
1x
2dx
x
1
)1x(x
Bx)1x(A
1x
B
x
A
xx
1x2 −
+−=−
+=−
+
Bx)1x(A1x +−=+
2B1x
1A0x
=⇒=−=⇒=
tdt2dx
tx
tx2
==
=
sentv
dxdu
tdtcosdv
tu
===
=
2
xv
dxx
1)xcos(lndu
xdxdv
)x(lnsenu
2
=
=
==
2
xv
dxx
1)x(lnsendu
xdxdv
)xcos(lnu
2
=
−=
==
12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K
21
x3
25
xdxx3dxx
dxx
3dx
x
xdx
x
3x2125
212322
Kx65
xx2Kx6
5
x2 25
++=++=
13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx
IsenxexcoseI xx −+−=
senxexcoseI2 xx +−=
( )∫ ++−= Ksenxexcose2
1dxsenxe xxx
14. ∫ ∫ ∫ =−=−= dxex2
3
2
exdxx3e
2
1
2
exdxex
x22x23
2x2x23x23
=
−−= ∫ dxxe
2
ex
2
3
2
ex x2x22x23
=
−+−= ∫ dx
2
e
2
xe
2
3
4
ex3
2
ex x2x2x22x23
K8
e3
4
xe3
4
ex3
2
ex x2x2x22x23
+−+−=
15. [ ]∫ ∫ ∫ +−=+++−=+
+−
=−
K1xln2
1K1xln1xln
2
1dx
1x
1
2
1dx
1x
1
2
1dx
1x
x 22
1x
)1x(B)1x(A
1x
B
1x
A
1x
x22 −
−++=+
+−
=−
)1x(B)1x(Ax −++=
2
1AA211xSi =⇒=⇒=
2
1BB211xSi =⇒−=−⇒−=
16. ( )∫ ∫∫∫∫∫ =+
++
−+=++−+−=
+−
dx1x
12dx
1x
x
x
2xlndx
1x
2xdx
x
12dx
x
1dx
1xx
2x222222
Karctgx21xln2
1
x
2xln 2 ++−−+=
xcosv
dxeu
dxsenxdv
eu
x
x
−==
==
senxv
dxedu
dxxcosdv
eu
x
x
==
==
2
ev
dxx3du
dxedv
xu
x2
2
x2
3
=
=
=
=
2
ev
xdx2du
dxedv
xu
x2
x2
2
=
==
=
2
ev
dxdu
dxedv
xu
x2
x2
=
==
=
