Lista de Derivadas-Integrales 16351
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A-1 DERIVADAS - INTEGRALES - 1 - LISTA (A) DE DERIVADAS 1. 0 [C] d dx = , C constante 2. 1 [ ] d x dx = , 2´. 0 , | | | | d x x dx x x = ≠ , 3. 1 n n [ ] n d x x dx − = 4. 1 2 [ ] d x dx x = 5. 3 2 3 1 3 [ ] ( ) d x dx x = 6. 1 4 3 4 4 [ ] ( ) d x dx x = 7. 2 1 1 [ ] d dx x x = − 8. 1 1 [ ] n n n d dx x x = − + 9. [ ] a a d dx x = , a constante 10. [ f( )] f( ) a a d x x dx ′ = 11. [f( ) g( )] f( ) g( ) d x x x x dx ′ ′ + = + 12. [f( ) g( )] f( ) g( ) d x x x x dx ′ ′ − = − 13. [af( ) b g( )] af( ) bg( ) d x x x x dx ′ ′ + = + , a y b son constantes . 14. [f( ) g( )] f( )g( ) f( )g( ) d x x x x x x dx ′ ′ ⋅ = + 15. 2 f( f ( )g( ) f( )g( ) g( ) [g( )] [ ] x x x x x d dx x x ′ ′ − = 16. [ ] e e x x d dx = 17. [ ] a a Ln(a) d dx x x = ⋅ , e Ln(a) Log (a) = , 0 a > , constante. 18. 1 Ln ( ) [ ] d x dx x = , 0 x > 2 DERIVADAS - INTEGRALES A - 1 19. 1 ( ) [ Ln a ] d dx x x = , a constante 0 ≠ 20. 1 Ln ( ) d x dx x − = , 0 x < 21. 1 a [ Log ( )] Ln(a) d x dx x = , a constante 0 > 22. FÓRMULAS DE CAMBIO DE BASE DE LOGARITMOS i) 10 Log (a) Log (a) ≡ , 0 a > ii) Ln (a) Log (a) e ≡ , 0 a > , 2 7182818 ... e . = iii) b Ln (a) Log (a) Log (a) Ln(b) Log (b) = = , 0 0 a , b > > iv) c b c Log (a) Log (a) Log (b) = , 0 0 0 a , b ,c > > > , 1 1 b , c ≠ ≠ . 23. [ Sen ( )] Cos ( ) d x x dx = 24. [ Cos ( )] Sen ( ) d x x dx = − 25. 2 [ Tan ( )] Sec ( ) d x x dx = 26. 2 [ Cot ( )] Cosec ( ) d x x dx = − 27. [ Sec ( )] Sec ( ) Tan ( ) d x x x dx = ⋅ 28. [ Cosec ( )] Cosec ( ) Cot ( ) d x x x dx = − ⋅ 29. 1 2 Senh ( ) ( ) e e x x x − = − , 1 2 Cosh ( ) ( ) e e x x x − = + 30. [ Senh ( )] Cosh ( ) d x x dx = , 31. [ Cosh ( )] Senh ( ) d x x dx = 31. 2 [ Tanh ( )] Sech ( ) d x x dx = , 32. 2 [ Coth ( )] Cosech ( ) d x x dx = − 33. [ Sech ( )] Sech ( ) Tanh ( ) d x x x dx = − 34. [ Cosech ( )] Cosech ( ) Coth ( ) d x x x dx = − .
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Lista de Integrales VENERO BALDEON.
Transcript of Lista de Derivadas-Integrales 16351
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A-1 DERIVADAS - INTEGRALES - 1 -
LISTA (A) DE DERIVADAS
1. 0[ C ]ddx
= , C constante
2. 1[ ]d xdx
= , 2. 0,| || |
d xx
dx
xx
= ,
3. 1n n[ ] nd
x x
dx
=
4. 1
2[ ]
dx
dx x
= 5. 3
231
3[ ]
( )
dx
dx x
=
6. 14
344[ ]
( )
dx
dx x
=
7. 2
1 1[ ]
d
dx x x
= 8. 1
1[ ]n nnd
dx x x
= +
9. [ ]a ad
dx
x = , a constante 10. [ f ( ) ] f ( )a ad
x x
dx
=
11. [ f ( ) g ( ) ] f ( ) g ( )d
x x x x
dx
+ = +
12. [ f ( ) g ( ) ] f ( ) g ( )d
x x x x
dx
=
13. [ a f ( ) b g( ) ] a f ( ) b g ( )d
x x x x
dx
+ = + , a y b son constantes .
14. [ f ( ) g ( ) ] f ( ) g ( ) f ( ) g ( )d
x x x x x x
dx
= +
15. 2
f ( f ( ) g ( ) f ( ) g ( )
g ( ) [ g ( ) ]
[ ]x x x x xd
dx x x
=
16. [ ]e ex xd
dx
=
17. [ ]a a Ln (a)d
dx
x x= , eLn ( a ) Log ( a )= , 0a > , constante.
18. 1
Ln ( )[ ]d
x
dx x
= , 0x >
2 DERIVADAS - INTEGRALES A - 1
19. 1
( )[ Ln a ]d
dx
xx
= , a constante 0
20. 1
Ln ( )d
x
dx x
= , 0x <
21. 1
a[ Log ( ) ]
Ln ( a )
dx
dx x
= , a constante 0>
22. FRMULAS DE CAMBIO DE BASE DE LOGARITMOS i) 10Log ( a ) Log ( a ) , 0a >
ii) Ln ( a ) Log ( a )e
, 0a > , 2 7182818 ...e .=
iii) b
Ln ( a ) Log ( a )Log ( a )
Ln ( b ) Log ( b)= = , 0 0a , b> >
iv) cb
c
Log ( a )Log ( a )
Log ( b )= , 0 0 0a , b , c> > > , 1 1b , c .
23. [ Sen ( ) ] Cos ( )d
x x
dx
= 24. [ Cos ( ) ] Sen ( )d
x x
dx
=
25. 2[ Tan ( ) ] Sec ( )d
x x
dx
= 26. 2[ Cot ( ) ] Cosec ( )d
x x
dx
=
27. [ Sec ( ) ] Sec ( ) Tan ( )d
x x x
dx
=
28. [ Cosec ( ) ] Cosec ( ) Cot ( )d
x x x
dx
=
29. 12
Senh ( ) ( )e exx
x
= , 12
Cosh ( ) ( )e exx
x
= +
30. [ Senh ( ) ] Cosh ( )d
x x
dx
= , 31. [ Cosh ( ) ] Senh ( )d
x x
dx
=
31. 2[ Tanh ( ) ] Sech ( )d
x x
dx
= , 32. 2[ Coth ( ) ] Cosech ( )d
x x
dx
=
33. [ Sech ( ) ] Sech ( ) Tanh ( )d
x x x
dx
=
34. [ Cosech ( ) ] Cosech ( ) Coth ( )d
x x x
dx
= .
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A-1 DERIVADAS - INTEGRALES - 3 -
35. 1
2
1
1Arc Sen ( ) Sen ( )
d dx x
dx dxx
= =
, 1 1,x
36. 12
1
1Arc Tan ( ) Tan ( )
d dx x
dx dx x
= =
+ , x
37. 2
1
1Arc Sec ( )
dx
dxx x
=
, 1x > .
38. 1
Arc Cot ( ) Arc Tan ( )xx
= ; 1 11
Cot ( ) Tan ( )xx
=
39. 1
Arc Sec ( ) Arc Cos ( )xx
= ; 1 11
Sec ( ) Cos ( )xx
= , 0x > .
40. 1
Arc Csc ( ) Arc Sen ( )xx
= ; 1 11
Csc ( ) Sen ( )xx
= , 0x > .
PROPIEDADES DE LOS L0GARITMOS
1. b
aLog ( ) b ax x= = , 0x > , 0a > , 1a
2. 10Log ( ) Log ( )x x= , eLn ( ) Log ( )x x= , 2 7182818.e =
3. 1 0a
Log ( ) = , 1a
Log ( a ) = , 1Ln (e ) =
4. ma a
Log ( ) m Log ( )x x= , mLn ( ) m Ln ( )x x=
5. a a a
Log ( MN ) Log (M) Log ( N )= +
6. a a a
MLog ( ) Log (M) Log ( N )
N=
7. a
Ln ( b ) Log ( b )Log ( b)
Ln (a ) Log (a )= =
8. FRMULA DE CAMBIO DE BASE DE LOGARITMOS
Si 0 1c , c> :
cac
Log ( |b )Log ( b )
Log ( a )=
9. Si A y B son nmeros positivos, 1A presentamos una forma de despejar x
en la ecuacin dada :
4 DERIVADAS - INTEGRALES A - 1
A B=x
Ln ( A ) Ln (B)=x Ln ( A ) Ln (B)=x de [4]
ALn (B)
Log (B)Ln ( A )
= =x
LISTA B : DERIVADAS COMPUESTAS
1. g ( ) f ( g ( ) ) g ( )f ( )d
x x x
dx
=
2. 1nn
[ f ( ) ] n [ f ( ) ] f ( ){ }d
x x x
dx
=
3. 2
f ( )f ( )
f ( )
xdx
dx x
= 4.
3233
f ( )f ( )
( f ( ) )
xdx
dx x
=
5. 21 f ( )
[ ]f ( ) [ f ( ) ]
xd
dx x x
=
6. f ( ) f ( )[ ] f ( )e ex xd
x
dx
=
7. f ( ) f ( )[ a ] a f ( ) Ln ( a )x xd
x
dx
= , a constante 0>
8. f ( )
Ln ( f ( )f ( )
xdx
dx x
=
9. a
f ( )Log ( f ( ) )
f ( ) Ln (a )
xdx
dx x
=
10. Sen [ f ( ) ] Cos [ f ( ) ] f ( )d
x x x
dx
=
11. Cos [ f ( ) ] Sen [ f ( ) ] f ( )d
x x x
dx
=
12. 2Tan[ f ( ) ] Sec [ f ( ) ] f ( )d
x x x
dx
=
13. 2Cot [ f ( ) ] Cosec [ f ( ) ] f ( )d
x x x
dx
=
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A-1 DERIVADAS - INTEGRALES - 5 -
14. Sec [ f ( ) ] Sec [ f ( ) ] Tan [ f ( ) ] f ( )d
x x x x
dx
=
15. Cosec [ f ( ) ] Cosec [ f ( ) ] Cot [ f ( ) ] f ( )d
x x x x
dx
=
35. 1
21
f ( )Arc Sen [ f ( ) ] Sen [ f ( ) ]
[ f ( ) ]
xd dx x
dx dxx
= =
,
36. 121
f ( )Arc Tan [ f ( ) ] Tan [ f ( ) ]
[ f ( ) ]
xd dx x
dx dx x
= =
+ ,
37. 2 1
f ( )Arc Sec [ f ( ) ]
[ f ( ) ] [ f ( ) ]
xdx
dxx x
=
38. f ( )
f ( ) f ( )f ( )
.| || |
xdx x
dx x
=
IDENTIDADES TRIGONOMTRICAS
1. 2 2 1Sen ( ) Cos ( )x x+ =
2. 2 21 Tan ( ) Sec ( )x x+ = 3) 2 21 Cot ( ) Cosec ( )x x+ =
4. 2 2 22 2 1Cos ( ) Cos ( ) Sen ( ) Cos ( )x x x x= =
21 2Sen ( )x= 5. 2 2Sen ( ) Sen ( ) Cos ( )x x x=
6. 2 1 1 22
Sen ( ) Cos ( )[ ]x x= , 21 1 22
Cos ( ) Cos ( )[ ]x x= +
7. 2
Sen ( ) Cos ( )x x= , 2
Cos Sen ( )x x=
(Esto ocurre con cada funcin trigonomtrica y su cofuncin)
8. Sen ( ) Sen ( )x x = Funcin Impar
9. Cos ( ) Cos ( )x x = Funcin Par
10. Tan ( ) Tan ( )x x = Funcin Impar
11. Sen ( )
Tan ( )Cos ( )
xx
x
= , ( )Cos
Ctg ( )Sen ( )
xx
x
=
12. 1
Sec ( )Cos ( )
x
x
= , 1
C0sec ( )Sen ( )
x
x
=
6 DERIVADAS - INTEGRALES A - 1
LISTA DE INTEGRALES
1. u Cdu = + , [ f ( ) ] f ( ) Cd x x= +
2. f ( ) f ( ) Cx dx x = +
3. 1 Cdx dx x= = + 4. Ca adx x= + , a constante.
5. 1
1
nn
Cn
xx dx
+= +
+ , n : entero, fraccin , ( ) , ( )+
6. 1
1 1
1n n( n ) Cdx
x x
= +
7. 21 1
Cdxxx
= +
8. 1
1
nn [ f ( ) ]
[ f ( ) ] f ( ) Cn
xx x dx
+ = +
+
9. 3 223
/Cx dx x= +
10. f ( )
Ln | f ( ) | Cf ( )
xdx x
x
= +
11. 2
1f ( )C
f ( )[ f ( ) ]
xdx
xx
= +
12. Ln ( ) Ln ( ) Cx dx x x x= +
13. Ce ex xdx = + 14. a
a CLn ( a )
xxdx = +
15. f ( ) f ( )f ( ) Ce ex xx dx = +
16. f ( )
f ( )f ( ) C
Ln ( a )
aa
xx
x dx = +
17. a f ( ) a f ( )x dx x dx= , a constante
f ( ) G ( ) Cx dx x= + si G ( ) f ( )x x =
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A-1 DERIVADAS - INTEGRALES - 7 -
18. [ f ( ) g ( ) ] f ( ) g ( )x x dx x dx x dx =
19. u dv uv v du = (INTEGRACIN POR PARTES)
20. b bb
aa af ( ) g ( ) f ( ) g ( ) g ( ) f ( )x x dx x x x x dx =
21. Sen ( ) Cos ( ) Cx dx x= +
22. Cos ( ) Sen ( ) Cx dx x= +
23. Tan ( ) Ln Cos ( ) C| |x dx x= + Ln Sec ( ) C| |x= +
24. Cot ( ) Ln Sen ( ) C| |x dx x= +
25. Sec ( ) Ln Sec ( ) Tan ( ) C| |x dx x x= + +
26. Cosec ( ) Ln Cosec ( ) Cot ( ) C| |x dx x x= +
27. 2 1 1 22
Sen ( ) Cos ( )[ ]x dx x dx= 21
2 2
Sen ( )[ ] C
xx= +
12
[ Sen ( ) Cos ( ) ] Cx x x= +
28. 2 1 1 22
Cos ( ) Cos ( )[ ]x dx x dx= + 21
2 2
Sen ( )[ ] C
xx= + +
12
[ Sen ( ) Cos ( ) ] Cx x x= + +
29. 2Sec ( ) Tan ( ) Cx dx x= +
30. 2Cosec ( ) Cot ( ) Cx dx x= +
31. 2 2 1Tan ( ) [ Sec ( ) ] Tan ( ) Cx dx x dx x x= = +
32. 2 2 1Cot ( ) Cosec ( ) Cot ( ) C[ ]x dx x dx x x= = +
33. 3 2Sen ( ) Sen ( ) Sen ( )x dx x x dx=
21[ Cos ( ) ] Sen ( )x x dx=
313
Cos ( ) Cos ( ) Cx x= + +
8 DERIVADAS - INTEGRALES A - 1
34. 3 313
Cos ( ) Sen ( ) Sen ( ) Cx dx x x= +
35. 3 2 1Tan ( ) Tan ( ) [ Sec ( ) ]x dx x x dx=
212
Tan ( ) Ln Cos ( ) C| |x x= + +
36. 3 212
Cot ( ) Cot ( ) Ln Sen ( ) C| |x dx x x= +
37. 3 12
Sec ( ) Sec ( ) Tan ( ) Ln Sec ( ) Tan ( ) C[ | | ]x dx x x x x= + + +
38. 3 12
C sc ( ) Cs c ( ) Cot ( ) Ln Cs c ( ) Cot ( ) C[ | | ]x dx x x x x= + +
38. 5 3 32 38 8
Tan ( )Sec ( ) Sec ( ) Sec ( ) Ln Sec Tan[ ] | |= + + +
xx dx x x x x
39. 5 3 32 3
8 8Cot ( )
Csc ( ) Csc ( ) Csc ( ) Ln Csc Cot[ ] | |x
x dx x x x x
= + +
39. Si f ( ) G ( ) Cx dx x= + , entonces G ( ) f ( )x x = ,
40. Sen [ g ( ) ] g ( ) Cos [ g ( ) ] Cx x dx x = +
41. Cos [ g ( ) ] g ( ) Sen [ g ( ) ] Cx x dx x = +
42. Tan [ g ( ) ] g ( ) Ln Cos [ g ( ) ] C| |x x dx x = +
43. Cot [ g ( ) ] g ( ) Ln Sen [ g ( ) ] C| |x x dx x = +
44. Sec [ g ( ) ] g ( ) Ln Sec [ g ( ) ] Tan [ g ( ) ] C| |x x dx x x = + +
45. Cosec [ g ( ) ] g ( ) Ln Cosec [ g ( ) ] Cot [ g( ) ] C| |x x dx x x = +
f ( h( ) ) h ( ) f ( h ( ) ) ( h ( ))
G ( h ( ) ) C
x x dx x d x
x
=
= +
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A-1 DERIVADAS - INTEGRALES - 9 -
46. 21
f ( )Arc Sen [ f ( ) ] C
[ f ( ) ]
x dxx
x
= +
47. 21
f ( )Arc Tan [ f ( ) ] C
[ f ( ) ]
x dxx
x
= +
+
48. 2 1
f ( )Arc Sec [ f ( ) ] C
f ( ) [ f ( ) ]
x dxx
x x
= +
, si 1f ( )x > .
49. 2 2
1
2
aLn C
a aa
| |xdxxx
= +
+ , 0a >
50. 2 2
1Arc Sen ( ) C
aa
xdx
x
= +
, 0a >
51. 2 2
1Arc Tan ( ) C
a aa
dx x
x
= ++
, 0a >
52. 2 2
1Arc Sec ( ) C
a aa
dx x
x x
= +
, si 0ax > >
53. 2 2
f ( ) f ( )Arc Sen [ ] C
[ f ( ) ]a
a
x dx x
x
= +
54. 2 21f ( ) f ( )
Arc Tan [ ] C
[ f ( ) ] a aa
x dx x
x
= +
+
55. 2 2
1f ( ) f ( )Arc Sec [ ] C
f ( ) [ f ( ) ]a a
a
x dx x
x x
= +
, 0f ( ) ax > > .
56. 2 2
12
f ( ) af ( )Ln C
a f ( ) a[ f ( ) ] a|| xx dx
xx
= +
+ , 0a >
57. 2 2
12
aLn C
a aa|| ud u
uu
= +
+ , 0a >
58. 2 2
1Arc Tan ( ) C
a aa
d u u
u
= ++
, 0a >
59. 2 2
Arc Sen ( ) Ca
a
d u udx
u
= +
, 0a >
