Z senn−1 x cos x n − 1 sen x dx = − senn−2 x dx + n n Z cosn−1 x sen x n − 1 n cos x dx = + cosn−2 x dx n n Z tann−1 x n tan x dx = − tann−2 x dx (n 6= 1) n−1 Z cotn−1 x n cot x dx = − cotn−2 x dx (n 6= 1) n−1 Z tan x secn−2 x n − 2 n + secn−2 x dx (n 6= 1) sec x dx = n−1 n−1 Z cot x csc n−1 x n − 2 n csc x dx = + cscn−2 x dx (n 6= 1) n−2 n−1 n
18. Z 19. Z 20. Z 21. Z 22. Z 23. Z 24.
senh x dx = cosh x Z
25.
cosh x dx = senh x
229 Z 26.
tanh x dx = log |cosh x| Z
27.
coth x dx = log |sen hx| Z
28.
sech x dx = arctan (senh x) Z
29. Z 30. Z 31.
¯ x ¯¯ 1 cosh x + 1 ¯ csch x dx = log ¯tanh ¯ = − 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
Z sech2 x dx = tanh x
32. Z
x x √ dx = xsenh−1 − x2 − a2 (a > 0) a a √ £ ¡ ¢ ¤ ½ Z xcosh−1 xa − √x2 − a2 £cosh−1 ¡ xa ¢ > 0, a > 0¤ −1 x 34. cosh dx = xcosh−1 xa + x2 − a2 cosh−1 xa < 0, a > 0 a Z ¯ ¯ x x a 35. tanh−1 dx = xtanh−1 + log ¯a2 − x2 ¯ a a 2 Z ³ ´ √ x 1 √ 36. dx = log x + a2 + x2 = sen h−1 (a > 0) a a2 + x2 Z 1 1 x 37. dx = arctan (a > 0) 2 2 a +x 2 a Z √ x x√ 2 a2 38. (a > 0) a2 − x2 dx = a − x2 + arcsen 2 2 a Z ¢√ ¡ 2 ¢3 x x¡ 2 3a4 5a − 2x2 arcsen (a > 0) 39. a − x2 2 dx = a 2 − x2 + 8 8 a Z 1 x √ 40. (a > 0) dx = arcsen a a2 − x2 ¯ ¯ Z ¯a + x¯ 1 1 ¯ dx = log ¯¯ 41. a2 − x2 2a a − x¯ 33.
senh−1
230
Tabla de Integrales Z
42.
1 (a2 − x2 )
Z √
3 2
dx =
a2
√
x a2 − x2
¯ ¯ √ x√ 2 a2 ¯ ¯ 2 2 2 43. ± dx = x ± a ± log ¯x + x ± a ¯ 2 2 Z ¯ ¯ √ x 1 ¯ ¯ √ dx = log ¯x + x2 − a2 ¯ = cosh−1 (a > 0) 44. a x2 − a 2 ¯ ¯ Z ¯ x ¯ 1 1 ¯ ¯ 45. dx = log ¯ x(a + bx) a a + bx ¯ Z
x2
a2
√
3
2 (3bx − 2a) (a + bx) 2 46. x a + bx dx = 15b2 Z Z √ √ 1 a + bx √ 47. dx = 2 a + bx + a dx x x a + bx √ Z x 2 (bx − 2a) a + bx √ 48. dx = 3b2 a + bx ¯√ ¯ ¯ a+bx−√ a¯ Z √1 log ¯ √ √ ¯ (a > 0) 1 a a+bx+ q a √ dx = 49. √2 arctan a+bx (a > 0) x a + bx −a −a Z √ 50. Z 51.
√ ¯ ¯ √ ¯ a + a2 − x2 ¯ a2 − x2 ¯ dx = a2 − x2 − a log ¯¯ ¯ x x
√ ¢3 1¡ x a2 − x2 dx = − a2 − x2 2 3
Z
√ ¢√ x¡ 2 a4 x x2 a2 − x2 dx = 2x − a2 a2 − x2 + arcsen 8 8 a √ ¯ ¯ Z ¯ a + a 2 − x2 ¯ 1 1 ¯ ¯ √ 53. dx = − log ¯ ¯ a x x a2 − x2 Z √ x √ 54. dx = − a2 − x2 a2 − x2 Z x2 x√ 2 x a2 √ 55. dx = − (a > 0) a − x2 + arcsen 2 2 2 2 a a −x ¯ ¯ Z √ 2 ¯ a + √x2 + a2 ¯ √ x + a2 ¯ ¯ 56. dx = x2 + a2 − a log ¯ ¯ ¯ ¯ x x 52.
(a > 0)
231 Z √ 57.
³x´ √ √ x2 − a 2 a dx = x2 − a2 − a arccos = x2 − a2 − arcsec x |x| a
(a > 0)
Z
√ ¢3 1¡ 2 x x2 ± a2 dx = x ± a2 2 3 ¯ ¯ Z ¯ ¯ x 1 1 ¯ √ √ 59. dx = log ¯¯ a x x2 + a2 a + x2 + a2 ¯ Z 1 1 a √ 60. dx = arccos (a > 0) a |x| x x2 − a 2 √ Z 1 x2 ± a2 √ 61. dx = ± a2 x x2 x2 ± a2 Z √ x √ 62. dx = x2 ± a2 x2 ± a 2 ¯ ¯ ( √ Z ¯ 2ax+b−√ b2 −4ac ¯ √ 1 log 1 ¯ ¯ (b2 > 4ac) b2 −4ac 2ax+b+ b2 −4ac 63. dx = ax2 + bx + c √ 2 arctan √2ax+b (b2 < 4ac) 4ac−b2 4ac−b2 Z Z ¯ 2 ¯ x 1 b 1 ¯ ¯ 64. dx = log ax + bx + c − dx 2 2 ax + bx + c 2a 2a ax + bx + c ( √ √ Z √1 log |2ax + b + 2 a ax2 + bx + c| (a > 0) 1 a √ dx = 65. −2ax−b √1 arcsen √ (a < 0) ax2 + bx + c −a b2 −4ac 58.
Z √
Z 2ax + b √ 2 4ac − b2 1 √ 66. + bx + c dx = ax + bx + c + dx 4a 8a ax2 + b + c √ Z Z x ax2 + bx + c b 1 √ √ 67. dx = − dx 2 2 a 2a ax + bx + c ax + bx + c ¯ √√ 2 ¯ ( Z ¯ 2 c ax +bx+c+bx+2c ¯ −1 √ log 1 ¯ ¯ (c > 0) x c √ dx = 68. bx+2c √1 arcsen √ (c < 0) x ax2 + bx + c −c |x| b2 −4ac ax2
µ
¶q 1 2 2 2 69. x + dx = x − a (a2 + x2 )3 5 15 q Z √ 2 2 ∓ (x2 ± a2 )3 x ±a dx = 70. x4 3a2 x3 Z sen(a − b)x sen(a + b)x 71. sen ax sen bx dx = − 2(a − b) 2(a + b) Z
3
√
x2
a2
¡
a2 6= b2
¢
232
Tabla de Integrales Z
72.
sen ax cos bx dx =
cos(a − b)x cos(a + b)x − 2(a − b) 2(a + b)
¡
cos ax cos bx dx =
sen(a − b)x sen(a + b)x − 2(a − b) 2(a + b)
¡
Z 73.
a2 6= b2
¢
a2 6= b2
¢
Z 74.
sec x tan x dx = sec x Z
75.
csc x cot x dx = −csc x Z
Z cosm−1 x senn−1 +x m − 1 + cosm−2 x senn x dx = cos x sen x dx = m+n m+nZ senn−1 x cosm+1 x n−1 = − + cosm x senn−2 x dx m+n m+n Z n 1 n n xn−1 cos ax dx x sen ax dx = − x cos ax + a a Z 1 n n n x cos ax dx = x sen ax − xn−1 sen ax dx a a Z xn eax n n ax x e dx = − xn−1 eax dx a a · ¸ 1 n n+1 log ax x log(ax) dx = x − n+1 (n + 1)2 Z xn+1 m m m n x (log ax) dx = (log ax) − xn (log ax)m−1 dx n+1 n+1 m
