Integrales

Anexo D
Tabla de Integrales
(PUEDE SUMARSE UNA CONSTANTE ARBITRARIA A CADA INTEGRAL)
1
xn+1
n+1
1. xn dx =
2. 1
dx = log | x |
x
3. ex dx = ex
4. ax dx =
5. sen x dx = − cos x
6. cos x dx = sen x
7. tan x dx = − log |cos x|
8. cot x dx = log |sen x|
9. sec x dx = log |sec x + tan x| = log tan
(n = −1)
ax
log a
227
1
1
x+ π
2
4
228
Tabla de Integrales
1
x
2
10.csc x dx = log |csc x − cot x| = log tan 11. arcsen x 12. arccos x 13. arctan x
x √ 2 x √ x a
dx = x arcsen dx = x arccos − a2 − x2 dx = x arctan − log a2 + x2
+ a − x2 a aa a a 2
a
14. sen2 mx dx = 1
(mx − sen mx cos mx)
2m
15. cos2 mx dx = 1
(mx + sen mx cos mx)
2m
16. sec2 x dx = tan x 17. csc2 x dx = −cot x 18. senn x dx = − 19. cosn x dx = cosn−1 x sen x n − 1 20. tann x dx = tann−1 x tann−2 x dx (n = 1)+ −
n n−1
n
21. cotn x dx = cotn−1 x cotn−2 x dx (n = 1)
−
n−1
22. secn x dx = tan x secn−2 x n −2 23. cot x csc n−1 x n − 2 24. senh x dx = cosh x 25. cosh x dx = senh x
+ csc x dx =
n−1 +
n−1 n−2
n−1
senn−1 x cos x n − 1
+
n
n
n
(a > 0)
(a > 0)
(a > 0)
senn−2 x dx
cosn−2 x dx
secn−2 x dx
cscn−2 x dx
(n = 1)
(n = 1)
229
26. tanh x dx = log |cosh x|
27. coth x dx = log |senhx|
28. sech x dx = arctan (senh x)
29. csch x dx = log tanh
30. 1
1
senh2 x dx = senh 2x − x
4
2
31. 1
1
cosh2 x dx = senh 2x + x
4
2
32. sech2 x dx = tanh x
33. senh−1
34.
x
x √
dx = xsenh−1 − x2 − a2 (a > 0)
a
a
√
xcosh−1 x − √x2 − a2 cosh−1
−1 x
a
cosh
dx =
xcosh−1 x + x2 − a2 cosh−1
a
a
35. tanh−1
36. √
37.
38.
39.
40.
41.
x
1
cosh x + 1
=− log
2
2
cosh x − 1
√
1
1
x
dx = arctan
2
+x
2
a
a2 − x2 dx =
a2 − x2
√
3
2
x
x√ 2
a2
a − x2 + arcsen
2
2
a
dx =
(a > 0)
(a > 0)
(a > 0)
√
x
x
3a4
5a2 − 2x2
arcsen
a 2 − x2 +
8
8
a
1
x
dx = arcsen
a
a2 − x2
a2
> 0, a > 0
< 0, a > 0
x
x a
dx = xtanh−1 + log a2 − x2
a
a 2
√
x
1
dx = log x + a2 + x2 = sen h−1
a
a2 + x2
a2
x
a
x
a
a+x
11
dx =
log
2
−x
2a
a−x
(a > 0)
(a > 0)
230
42.
43.
Tabla de Integrales
1
(a2 − x2 )
√
x2
±
a2
3
2
dx =
a2
√
x
a2 − x2
√
x√ 2
a2
2±
dx =
x ±a
log x + x2 ± a2
2
2
√
x
1
dx = log x + x2 − a2 = cosh−1
a
x2 − a 2
44. √
45. x
1
1
dx = log
x(a + bx)
a
a + bx
46.
47.
48.
49.
(a > 0)
3
√
2 (3bx − 2a) (a + bx) 2
x a + bx dx =
15b2
√
√
1
a+ bx
√
dx = 2 a + bx + a
dx
x
x a + bx
√
x
2 (bx − 2a) a + bx
√
dx =
3b2
a + bx

√
√
1
 √ log √a+bx−√a (a > 0)
1
a
a+bx+ a
√
dx =
 √2 arctan a+bx (a > 0)
x a + bx
−a
−a
√
50.
51.
52.
53.
√
1
x a2 − x2 dx = − a2 − x2
3
√
a2 − x2
x
3
2
√
√
x
a4
x
x2 a2 − x2 dx =
2x2 − a2
a2 − x2 + arcsen
8
8
a
√
a + a 2 − x2
1
1
√
dx = − log
a
x
x a2 −x2
54. √
55. √
56.
√
a2 − x2
a+
dx = a2 − x2 − a log
x
√
x
dx = − a2 − x2
a2 − x2
x2
x√ 2
x
a2
dx = −
(a > 0)
a − x2 + arcsen
2 − x2
2
2
a
a
√
√
√
a + x2 + a2
x2 + a2
dx = x2 + a2 − a log
x
x
(a > 0)
231
√
57.
58.
59.
60.
61.
√
√
x2 − a 2
a
x
dx = x2 − a2 − a arccos
= x2 − a2 − arcsec
x
|x|
a
√
1 2
x x2 ± a2 dx =
x ± a2
3
x
√
1
x2
+
a2...