1. This document provides a summary of basic integrals and trigonometric identities relevant for calculus. It lists 20 basic integrals involving functions like x, 1/x, e^x, sin(x), and cotan(x). 2. It also covers integral substitution, integration by parts, and trigonometric identities like sin^2(x) + cos^2(x) = 1. 3. Additional trigonometric identities are presented for functions like sin(x ± y), cos(x ± y), and tan(x ± y) in terms of sums and differences of trigonometric functions.

1. Universidad Nacional Andr´es Bello
Departamento de Matem´aticas
C´alculo Integral
Profesor Javier Olivos
Resumen de Integrales. Autor: Mauricio Vargas
Integrales B´asicas
1. dx = x + c
2. kdx = kx + c (k cte)
3. xn
dx =
xn+1
n + 1
+ c, n = −1
4.
1
x
dx = ln(|x|) + c
5. eax
dx =
eax
a
+ c
6. abx
dx =
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 + x2
dx =
1
a
arctan
x
a
+ c
18.
1
a2 − x2
dx =
−1
2a
ln
x − a
x + a
+ c
19.
1
x2 − a2
dx =
1
2a
ln
x − a
x + a
+ c
Sustituci´on
g(f(x)) · f (x)dx = g(u)du el cambio de variable es u = f(x)
Integraci´on por partes
udv = uv − vdu
Identidades trigonom´etricas
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)]