1. Vidal Matias Marca Facultad de Ciencias y Tecnolog´ıa 1
Tabla de Integrales
Formas Elementales
1. xn
dx =
xn+1
n + 1
+ C n = −1
2.
1
x
dx = ln |x| + C
3. ex
dx = ex
+ C
4. ax
dx =
ax
ln a
+ C
5. sin xdx = − cos x + C
6. cos xdx = sin x + C
7. sec2
xdx = tan x + C
8. csc2
xdx = − cot x + C
9. sec x tan xdx = sec x + C
10. csc x cot xdx = − cscx + C
11. tan xdx = − ln | cos x| + C
12. cot xdx = ln | sin x| + C
13. sec xdx = ln | sec x + tan x| + C
14. csc xdx = ln | csc x − cot x| + C
15.
1
√
a2 − x2
dx = arcsin
x
a
+ C
16.
1
a2 + x2
dx =
1
a
arctan
x
a
+ C
17.
1
a2 − x2
dx =
1
2a
ln
a + x
a − x
+ C
18.
1
x
√
x2 − a2
dx =
1
a
arcsec
x
a
+ C
Formas Trigonom´etricas
19. sin2
xdx =
1
2
u −
1
4
sin 2x + C
20. cos2
xdx =
1
2
u +
1
4
sin 2x + C
21. tan2
xdx = tan x − x + C
22. cot2
xdx = − cot x − x + C
23. sin3
xdx = −
1
3
(2 + sin2
x) cos x + C
24. cos3
xdx =
1
3
(2 + cos2
x) sin x + C
25. tan3
xdx =
1
2
tan2
x + ln | cos x| + C
26. cot3
xdx = −
1
2
cot2
x − ln | sin x| + C
27. sec3
xdx =
sec x tan x + ln | sec x + tan x|
2
+ C
28. csc3
xdx =
− csc x cot x + ln | csc x − cot x|
2
+ C
29. sin ax sin bxdx =
sin(a − b)x
2(a − b)
−
sin(a + b)x
2(a + b)
+ C a2
= b2
30. cos ax cos bxdx =
sin(a − b)x
2(a − b)
+
sin(a + b)x
2(a + b)
+ C a2
= b2
31. sin ax cos bxdx = −
cos(a − b)x
2(a − b)
−
cos(a + b)x
2(a + b)
+ C a2
= b2
32. sinn
xdx = −
sinn−1
x cos x
n
+
n − 1
n
sinn−2
xdx
33. cosn
xdx =
cosn−1
x sin x
n
+
n − 1
n
cosn−2
xdx
34. tann
xdx =
tann−1
x
n − 1
− tann−2
xdx n = 1
35. cotn
xdx =
cotn−1
x
1 − n
− cotn−2
xdx n = 1
36. secn
xdx =
secn−2
x tan x
n − 1
+
n − 2
n − 1
secn−2
xdx n = 1
37. cscn
xdx =
cscn−2
x cot x
1 − n
+
n − 2
n − 1
cscn−2
xdx n = 1
38. xn
sin xdx = −xn
cos x + n xn−1
cos xdx 39. xn
cos xdx = xn
sin x − n xn−1
sin xdx
40. sinn
x cosm
xdx = −
sinn−1
x cosm+1
x
n + m
+
n − 1
n + m
sinn−2
x cosm
xdx n = −m
41. sinn
x cosm
xdx =
sinn+1
x cosm−1
x
n + m
+
m − 1
n + m
sinn
x cosm−2
xdx n = −m
2. Vidal Matias Marca Facultad de Ciencias y Tecnolog´ıa 2
Formas que Incluyen x2 ± a2, a2 − x2
42. x2 ± a2dx =
x
√
x2 ± a2
2
±
a2
ln |x +
√
x2 ± a2|
2
+ C
43. a2 − x2dx =
x
√
a2 − x2
2
+
a2
2
arcsin
x
a
+ C
44.
√
x2 + a2
x
dx = x2 + a2 − a ln
a +
√
x2 + a2
x
+ C
45.
√
x2 − a2
x
dx = x2 − a2 − a arcsin
x
a
+ C
46.
√
a2 − x2
x
dx = a2 − x2 − a ln
a +
√
a2 − x2
x
+ C
47.
1
√
x2 ± a2
dx = ln |x + x2 ± a2| + C
48.
1
x2
√
x2 ± a2
dx = ∓
√
x2 ± a2
a2x
+ C
49.
1
x2
√
a2 − x2
dx = −
√
a2 − x2
a2x
+ C
50.
1
x
√
a2 − x2
dx = −
1
a
ln
a +
√
a2 − x2
x
+ C
51.
x2
√
x2 ± a2
dx =
x
√
x2 ± a2
2
∓
a2
ln |x +
√
x2 ± a2|
2
+ C
52.
√
x2 ± a2
x2
dx = −
√
x2 ± a2
x
+ ln |x + x2 ± a2| + C
53.
x2
√
a2 − x2
dx = −
x
√
a2 − x2
2
+
a2
2
arcsin
x
a
+ C
54.
√
a2 − x2
x2
dx = −
√
a2 − x2
x
− arcsin
x
a
+ C
Formas Exponenciales y Logar´ıtmicas
55. xn
ex
dx = xn
ex
− n xn−1
ex
dx
56. xn
ln xdx =
xn+1
n + 1
ln x −
xn+1
(n + 1)2
+ C
57. eax
sin bxdx =
eax
a2 + b2
(a sin bx − b cos bx) + C
58. eax
cos bxdx =
eax
a2 + b2
(a cos bx + b sin bx) + C
Integraci´on por Partes
udv = uv − vdu
Integraci´on por Sustituci´on Tri-
gonom´etricas
x2 + a2 ⇒ x = a tan θ
x2 − a2 ⇒ x = a sec θ
a2 − x2 ⇒ x = a sin θ
Fracciones Parciales
pm(x)
(x − a1)(x − a2)...(x − an)
=
A
x − a1
+
B
x − a2
+ ... +
N
x − an
pm(x)
(x − a1)n
=
A
x − a1
+
B
(x − a1)2
+ ... +
N
(x − a1)n
pm(x)
(x2 + b1x + c1)(x2 + b2x + c2)...(x2 + bnx + cn)
=
A1x + B1
x2 + b1x + c1
+
A2x + B2
x2 + b2x + c2
+ ... +
Anx + Bn
x2 + bnx + cn
pm(x)
(x2 + b1x + c1)n
=
A1x + B1
x2 + b1x + c1
+
A2x + B2
(x2 + b1x + c1)2
+ ... +
Anx + Bn
(x2 + b1x + c1)n
C´alculo de ´Areas por Integraci´on
A
a b
f(x)
g(x)
A =
b
a
f(x) − g(x) dx
A
c
d
f(y)
g(y)
A =
d
c
f(y) − g(y) dy
