Universidad Nacional Andres BelloDepartamento de Matematicas
Calculo IntegralProfesor Javier Olivos
Resumen de Integrales. Autor: Mauricio Vargas
Integrales Basicas
1.
∫dx = x + c
2.
∫kdx = kx + c (k cte)
3.
∫xndx =
xn+1
n + 1+ c, n 6= −1
4.
∫1
xdx = ln(|x|) + c
5.
∫eaxdx =
eax
a+ c
6.
∫abxdx =
abx
ln(a) · b+ c, a > 0
7.
∫sen(x)dx = − cos(x)dx + c
8.
∫cos(x)dx = sen(x) + c
9.
∫tan(x)dx = ln | sec(x)|+ c
10.
∫cotan(x)dx = ln | sen(x)|+ c
11.
∫sec(x)dx = ln | sec(x) + tan(x)|+ c
12.
∫cosec(x) = ln | cosec(x)− cotan(x)|+ c
13.
∫sec2(x)dx = tan(x) + c
14.
∫cosec2(x)dx = −cotan(x) + c
15.
∫sec(x) tan(x)dx = sec(x) + c
16.
∫cosec(x)cotan(x)dx = − cosec(x) + c
17.
∫1
a2 + x2dx =
1
aarctan
(xa
)+ c
18.
∫1
a2 − x2dx =
−1
2aln
∣∣∣∣x− a
x + a
∣∣∣∣+ c
19.
∫1
x2 − a2dx =
1
2aln
∣∣∣∣x− a
x + a
∣∣∣∣+ c
Sustitucion ∫g(f(x)) · f ′(x)dx =
∫g(u)du el cambio de variable es u = f(x)
Integracion por partes ∫udv = uv −
∫vdu
Identidades trigonometricas
1. sen2(x) + cos2(x) = 1
2. tan(x) =sen(x)
cos(x)
3. cotan(x) =cos(x)
sen(x)
4. sec(x) =1
cos(x)
5. cosec(x) =1
sec(x)
6. 1 + tan2(x) = sec2(x)7. 1 + cotan2(x) = cosec2(x)8. sen(2x) = 2 sen(x) cos(x)9. cos(2x) = cos2(x)− sen2(x)
10. tan(2x) =2 tan(x)
1− tan2(x)
11. sen2(x) =1− cos(2x)
2
12. cos2(x) =1 + cos(2x)
2
13. tan2(x) =1− cos(2x)
1 + cos(2x)
14. sen(2x) =2 tan(x)
1 + tan2(x)
15. cos(2x) =1− tan2(x)
1 + tan2(x)
Identidades adicionales
1. sen(x± y) = sen(x) cos(y)± sen(y) cos(x)2. cos(x± y) = cos(x) cos(y)∓ sen(y) sen(x)
3. tan(x± y) =tan(x)± tan(y)
1∓ tan(x) tan(y)
4. sen(x) sen(y) =1
2[cos(x− y)− cos(x + y)]
5. sen(x) cos(y) =1
2[sen(x + y) + sen(x− y)]
6. cos(x) cos(y) =1
2[cos(x + y) + cos(x− y)]