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
1x+ π
2
4

228

Tabla de Integrales
1
x
2

10.

csc x dx = log |csc x − cot x| = log tan

11.

arcsen

x
x √2
dx = x arcsen
+ a − x2
a
a

12.

arccos

x
x√
dx = x arccos − a2 − x2
a
a

13.

arctan

x
xa
dx = x arctan − log a2 + x2
a
a2

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
+
n
n

20.

tann x dx =

tann−1 x
−
n−1

tann−2 x dx

21.

cotn x dx =

cotn−1 x
−
n−1

cotn−2 x dx

22.

secn x dx =

tan x secn−2 x n − 2
+
n−1
n−1

23.

cot x csc n−1 x n − 2
csc x dx =
+
n−2
n−1

24.

senh x dx = cosh x

25.

cosh xdx = senh x

senn−1 x cos x n − 1
+
n
n

n

(a > 0)
(a > 0)
(a > 0)

senn−2 x dx
cosn−2 x dx
(n = 1)
(n = 1)
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 |sen hx|

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 = senh2x + 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
a2a − 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
xa
dx = xtanh−1 + log a2 − x2
a
a2

√
x
1
dx = log x + a2 + x2 = sen h−1
a
a2 + x2

a2

x
a
x
a

a+x
1
1
dx =
log
2
−x
2a
a−x

(a > 0)

(a > 0)

230

42.

43.

Tabla deIntegrales
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
xa + 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

√
12
x x2 ± a2 dx =
x ± a2
3
x

√

1
x2

+a2

dx =

x
1
√
log
a
a + x2 + a2

1

√

63.

1
dx =
2 + bx + c
ax

65.

66.
67.

(a > 0)

√
x
dx = x2 ± a2
x2 ± a 2

62.

64.

3
2

1
a
dx = arccos
a
|x|
x x2 − a 2
√
1
x2 ± a2
√
dx = ±
a2 x
x2 x2 ± a2
√

(a > 0)

√
b− 2
√1
log 2ax+b+√b2 −4ac
b2 −4ac
2ax+
b −4ac
2
√2
√ ax+b
arctan 4ac−b2
4ac−b2

(b2 > 4ac)
(b2 < 4ac)...