Tabla de integrales

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Anexo D


(PUEDE SUMARSE UNA CONSTANTE ARBITRARIA A CADA INTEGRAL)

1. 2. 3. 4. 5. 6. 7. 8. 9.

xn dx =

1 xn+1 n+1

(n = −1)

1 dx = log | x | x ex dx = ex ax dx = ax log a

sen x dx = − cos x cos x dx = sen x tan x dx = − log |cos x| cot x dx = log |sen x| sec x dx = log |sec x + tan x| = log tan 227 1 1 x+ π 2 4

228 1 x 2 (a > 0) (a > 0) (a > 0)

Tabla de Integrales

10.11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

csc x dx = log |csc x − cot x| = log tan arcsen arccos arctan x x √ 2 dx = x arcsen + a − x2 a a x x √ dx = x arccos − a2 − x2 a a

x x a dx = x arctan − log a2 + x2 a a 2 1 (mx − sen mx cos mx) 2m 1 (mx + sen mx cos mx) 2m

sen2 mx dx = cos2 mx dx =

sec2 x dx = tan x csc2 x dx = −cot x senn x dx = − cosn x dx = tann x dx =cotn x dx = secn x dx =
n

senn−1 x cos x n − 1 + n n

senn−2 x dx cosn−2 x dx (n = 1) (n = 1) secn−2 x dx cscn−2 x dx (n = 1) (n = 1)

cosn−1 x sen x n − 1 + n n tann−1 x − n−1 cotn−1 x − n−1 tann−2 x dx cotn−2 x dx

tan x secn−2 x n − 2 + n−1 n−1

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

229

26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.38. 39. 40. 41.

tanh x dx = log |cosh x| coth x dx = log |sen hx| sech x dx = arctan (senh x) csch x dx = log tanh x 1 cosh x + 1 = − log 2 2 cosh x − 1

1 1 senh2 x dx = senh 2x − x 4 2 1 1 cosh2 x dx = senh 2x + x 4 2 sech2 x dx = tanh x senh−1 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 tanh−1 √ x x a dx = xtanh−1 +log a2 − x2 a a 2 (a > 0)

x a x a

> 0, a > 0 < 0, a > 0

√ x 1 dx = log x + a2 + x2 = sen h−1 a a2 + x2 1 1 x dx = arctan 2 +x 2 a (a > 0)

a2 √

a2 − x2 dx =
3 2

x x√ 2 a2 a − x2 + arcsen 2 2 a

(a > 0) (a > 0)

a2 − x2 √

dx =

√ x x 3a4 5a2 − 2x2 arcsen a 2 − x2 + 8 8 a (a > 0)

1 x dx = arcsen a a2 − x2 a+x 1 1 dx = log 2 −x 2a a−x

a2

230 1 (a2 − x2 ) √ x2 ± a23 2

Tabla de Integrales x a2 − x2

42.

dx =

a2



43. 44. 45.

√ x√ 2 a2 2± dx = x ±a log x + x2 ± a2 2 2 (a > 0)



√ x 1 dx = log x + x2 − a2 = cosh−1 a x2 − a 2

x 1 1 dx = log x(a + bx) a a + bx 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

46. 47. 48.



3

49.

√ 50. 51. 52. 53. 54. 55.

√ a2 − x2 a+ dx = a2 − x2 − a log x
3 2



a2 − x2 x

√ 1 x a2 − x2 dx = − a2 − x2 3

√ √ 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 √ √ √ x dx = − a2 − x2 a2 − x2

(a > 0)

56.

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

231 √ 57. 58. 59. 60. 61. 62. √ √ x2 − a 2 a x dx = x2 − a2 − a arccos = x2 − a2 − arcsec x |x| a
3 2

(a > 0)

√ 1 2 x x2 ± a2 dx = x ± a2 3 x √ √ 1 x2 1 + a2 dx =

x 1 √ log a a + x2 + a2 (a > 0)

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

63.

1 dx= 2 + bx + c ax

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

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

64. 65.

x 1 b 1 dx = log ax2 + bx + c − dx 2 + bx + c + bx + c 2a 2a ax √ √ 1 √ log |2ax + b + 2 a ax2 + bx + c| (a > 0) 1 a √ dx = −2ax−b √1 arcsen √ (a < 0) ax2 + bx + c −a b2 −4ac ax2 √ 2ax + b √ 2 4ac − b2 1 √ ax + bx + c + dx 4a 8a ax2 + b + c √ x ax2 + bx + c b1 √ √ dx = − dx 2 + bx + c 2 + bx + c a 2a ax ax ax2 + bx + c dx = 1 dx = 2 + bx + c x ax √ √ x3 x2 + a2 dx = √ x2 ± a 2 dx = x4
√ √ 2 c ax2 +bx+c+bx+2c −1 √ log x c bx+2c √1 arcsen √ −c |x| b2 −4ac

66. 67.

68.

(c > 0) (c < 0)

69.

1 2 2 x − a2 5 15 (x2 ± a2 )3 3a2 x3

(a2 + x2 )3

70. 71.

sen ax sen bx dx =

sen(a − b)x sen(a + b)x − 2(a − b) 2(a + b)

a2 = b2

232...
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