Integrales
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En este document encontrarás toda la metodología para resolver cualquier tipo de integ...
www.fisicaeingenieria.es Tabla de integrales
1
Tipos FormasFormasFormasFormas
SimpleSimpleSimpleSimple CompuestaCompuestaC...
www.fisicaeingenieria.es Tabla de integrales
2
TTTTipo potencialipo potencialipo potencialipo potencial ( )1a ≠ −
0dx C=∫ ...
www.fisicaeingenieria.es Tabla de integrales
3
1
dx L x
x
=∫
f
dx L f
f
′
=∫
Ejercicios resueltos:Ejercicios resueltos:Eje...
www.fisicaeingenieria.es Tabla de integrales
4
Tipo senoTipo senoTipo senoTipo seno
s n cose xdx x=−∫
s n cose f f dx f′⋅ ...
www.fisicaeingenieria.es Tabla de integrales
5
Tipo tangenteTipo tangenteTipo tangenteTipo tangente
2
sec tanxdx x=∫
2
sec...
www.fisicaeingenieria.es Tabla de integrales
6
Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno)Tipo arco seno (=...
www.fisicaeingenieria.es Tabla de integrales
7
55.55.55.55.
( )
2
24 2
1 2 1
arctg
1 2 21
x x
dx dx x
x x
= =
+ +
∫ ∫
56.5...
www.fisicaeingenieria.es Tabla de integrales
8
6.6.6.6.
( )
( ) ( )
( )
4 1
4 4
4
2 11 1 1
2 1 2 1 2
2 2 4 12 1
x
dx x dx ...
www.fisicaeingenieria.es Tabla de integrales
9
2
2
1 2 2
1 2
2
cos x sen x
cos x
sen x
− =
−
=
13.13.13.13. ( ) ( )2 2 2
1...
www.fisicaeingenieria.es Tabla de integrales
10
20.20.20.20. ( )
3
2 23
2 2 2 2 2
2
1
1 1 2 1
31 2
xx
dx x x x x dx x x C
...
www.fisicaeingenieria.es Tabla de integrales
11
25.25.25.25. 2 2 22
1 1
1 1 1 1 14 22
44 4 4 2 2
1 1
4 4 2 2
x
dx dx dx dx...
www.fisicaeingenieria.es Tabla de integrales
12
Resumiendo: 2
2
x x x x
x x x
x x
x
e cosx dx e cosx e senx e cosx dx
e co...
www.fisicaeingenieria.es Tabla de integrales
13
39.39.39.39.
( ) ( ) 22
1 1 1
2 2 2 2
111
dx t dt dt arctgt C arctg x C
tt...
www.fisicaeingenieria.es Tabla de integrales
14
1 0 0 1
1
1 1 1
0
−
1 1 1
( )( )2 2
1 1 1t t t t− = − + +
1 1 1 0 0 0 0
1
...
www.fisicaeingenieria.es Tabla de integrales
15
2
1
2 1 2 1
1 2
t
t dt t ln t C
t
  
= − + = − + + + =  
+   
...
www.fisicaeingenieria.es Tabla de integrales
16
2 2
2 2
2
1
cos x cos x sen x
cos x sen x
= −
= +
2
2
1 2 2
1 2
2
cos x co...
www.fisicaeingenieria.es Tabla de integrales
17
60.60.60.60.
2 3
2 1
2 2 2 1
1 1 11
x t t
dx t dt dt t t dt
t t tx
 
= ⋅...
www.fisicaeingenieria.es Tabla de integrales
18
1 1 1
3
9 3 9
x
x
x ln e C
e
= − + + − +
63.63.63.63.
( ) ( )
( ) 22
1 1 1...
www.fisicaeingenieria.es Tabla de integrales
19
4
4 2
2
2
3
3 12
12
48
x
x x
x
x
− +
−12 +
48
( ) ( )
( )( )
( ) ( )
2
2
4...
www.fisicaeingenieria.es Tabla de integrales
20
( ) ( ) 2
2 1 1 1
0 1 1
1 3 3
2 5 2 4 5 2 1 12 6 2 3
x Ax x B x Cx
x B B
x...
www.fisicaeingenieria.es Tabla de integrales
21
2
2 2
1 2 2 2
2 2 2 2 2
sen t sent cost
cos t t t
cos t dt dt C C
⋅
+
= ⋅ ...
www.fisicaeingenieria.es Tabla de integrales
22
1 2 1 4 2 4
2 8 2 4 8 32
cos x cos x x sen x x sen x
dx C
− − 
= − = − −...
www.fisicaeingenieria.es Tabla de integrales
23
( )
2
3 3 3
3
1 3
x x
x senx ln cosx
cosx dx C
ln
+ ⋅
⋅ ⋅ = +
+
∫
82.82.82...
www.fisicaeingenieria.es Tabla de integrales
24
1 1 2 0
0
0 0 0
1 1 2 0
1
1 2
1 2 0
2
2
0
−
−
−
−
1
( )( )
( )( ) ( ) ( )
...
www.fisicaeingenieria.es Tabla de integrales
25
89.89.89.89.
2 2
2 2 2 2 2 2 2
2
a sen t a a
a b x dx a b cost dt a a sen ...
www.fisicaeingenieria.es Tabla de integrales
26
1 41 11 1
3 32 21 11 1
3 32 2
5 4 7 5 4 7
1 1 4 11 1
3 2 3 2
x x x x
C C
+...
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  1. 1. Integrales www.fisicaeingenieria.es En este document encontrarás toda la metodología para resolver cualquier tipo de integral definido o indefinida Luis Muñoz Mato
  2. 2. www.fisicaeingenieria.es Tabla de integrales 1 Tipos FormasFormasFormasFormas SimpleSimpleSimpleSimple CompuestaCompuestaCompuestaCompuesta Tipo potencial a ≠ -1 ∫ + = + 1 1 a x dxx a a ∫ + =′⋅ + 1 1 a f dxff a a 4 51 5 x dx x=∫ ( )( ) ( ) 312 302 1 2 1 1 31 x x x x x dx + + + + + =∫ Tipo logarítmico xLdx x =∫ 1 fLdx f f = ′ ∫ 3 1 3 3dx dx L x x x = =∫ ∫ 2 3 3 3 1 8 8 3 = + +∫ x dx L x x Tipo exponencial xx edxe =∫ La a dxa x x =∫ ∫ ′=′⋅ edxfe f La a dxfa f ′ =′⋅∫ 2 1 2 1 2 11 1 2 2 x x x e dx e dx e+ + + = ⋅ =∫ ∫ 2 1 2 1 2 11 1 3 3 2 2 ln 3 + + + = ⋅ =∫ ∫ x x x e dx dx Tipo seno ∫ = senxxdxcos ∫ =′⋅ senfdxffcos 1 1 s n 2 s n 2 2 cos 2 2 2 e x dx e x dx x= ⋅ = −∫ ∫ Tipo coseno ∫ −= xsenxdx cos ∫ −=′⋅ fdxfsenf cos ( ) ( ) ( )2 2 2 1 cos 1 s n 1x x x dx e x x+ ⋅ + + = + +∫ Tipo tangente tgxxdx =∫ 2 sec ( ) tgxdxxtg =+∫ 2 1 tgxdx x =∫ 2 cos 1 tgfdxff =′⋅∫ 2 sec ( ) tgfdxfftg =′⋅+∫ 2 1 tgfdx f f = ′ ∫ 2 cos 2 2 3sec 3 sec 3tanx dx x dx x= =∫ ∫ 2 2 7 7 sec 7 tan cos = =∫ ∫dx x dx x x ( ) ( )2 2 5 5tan 5 1 tan 5tanx dx x dx x+ = + =∫ ∫ Tipo arco seno arcsenxdx x = − ∫ 2 1 1 a x arcsendx xa = − ∫ 22 1 arcsenfdx f f = − ′ ∫ 2 1 a f arcsendx fa f = − ′ ∫ 22 ( ) 2 4 22 2 2 arcsen 1 1 x x dx dx x x x = = − − ∫ ∫ Tipo arco tangente arctgxdx x = +∫ 2 1 1 a x arctg a dx xa 11 22 = +∫ arctgfdx f f = + ′ ∫ 2 1 a f arctg a dx fa f 1 22 = + ′ ∫ 2 2 1 1 1 1 arctg 3 3 3 1 3 dx dx x x x = = + +∫ ∫ TTTT
  3. 3. www.fisicaeingenieria.es Tabla de integrales 2 TTTTipo potencialipo potencialipo potencialipo potencial ( )1a ≠ − 0dx C=∫ k dx kx C= +∫ 1 1 a a x x dx a + = +∫ 1 1 a a f f f dx a + ′⋅ = +∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 1.1.1.1. 4 51 5 x dx x=∫ 2.2.2.2. 3 3 4 4 1 3 3 x x dx x dx x − − − = = = − −∫ ∫ 3.3.3.3. 5 2 53 3 3 3 5 5 3 x x dx x= =∫ 4.4.4.4. 2 1 23 3 3 3 2 2 3 x x dx x − = =∫ 5.5.5.5. ( ) ( ) 2 31 1 1 3 x dx x+ = +∫ 6.6.6.6. ( )( ) ( ) 312 302 1 2 1 1 31 x x x x x dx + + + + + =∫ 7.7.7.7. 3 41 s n cos s n 4 e x x dx e x=∫ 8.8.8.8. ( ) 2 2 31 sec 3 tg x x dx tg x=∫ 9.9.9.9. ( ) ( )3 5 3 2 41 1 4 tg x tg x dx tg x tg x dx tg x+ = + =∫ ∫ 10.10.10.10. ( )3 3 2 4 31 1 cos 1 s n sin 4 3 xdx tg x tg x dx tg x e x x= + = = −∫ ∫ 11.11.11.11. ( ) ( )3 2 2 31 s n 1 cos s n cos s n cos cos 3 sen xdx e x x dx e x e x dx x x= − = − = − +∫ ∫ ∫ Tipo logarítmicoTipo logarítmicoTipo logarítmicoTipo logarítmico
  4. 4. www.fisicaeingenieria.es Tabla de integrales 3 1 dx L x x =∫ f dx L f f ′ =∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 1.1.1.1. 3 1 3 3dx dx L x x x = =∫ ∫ 2.2.2.2. 2 3 3 3 1 5 5 x dx L x x x x + = + + + +∫ 3.3.3.3. ( )2 2 2 1 2 1 1 1 2 1 2 x x dx dx L x x x = = + + +∫ ∫ 4.4.4.4. 2 2 3 3 3 1 3 1 8 8 3 8 3 x x dx dx L x x x = = + + +∫ ∫ 5.5.5.5. s n cos cos e x tg x dx dx Ln x x = = −∫ ∫ 6.6.6.6. cos cotg s n s n x x dx dx L e x e x = =∫ ∫ Tipo exponencialTipo exponencialTipo exponencialTipo exponencial x x e dx e=∫ x x a a dx La =∫ f f e f dx e′⋅ =∫ f f a a f dx La ′⋅ =∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 12.12.12.12. 2 1 2 1 2 11 1 2 2 x x x e dx e dx e+ + + = ⋅ =∫ ∫ 13.13.13.13. 3 3 3 x x dx L =∫ 14.14.14.14. 3 3 3 2 32 2 2 x xx x dx dx L        = =         ∫ ∫ 15.15.15.15. 2 2 21 1 2 2 2 x x x x e dx x e dx e= ⋅ =∫ ∫ 16.16.16.16. sin sin cosx x e x dx e=∫ 17.17.17.17. 2 2 s n s n 2s n cos s ne x e x e e x x dx e e x dx e= =∫ ∫
  5. 5. www.fisicaeingenieria.es Tabla de integrales 4 Tipo senoTipo senoTipo senoTipo seno s n cose xdx x=−∫ s n cose f f dx f′⋅ = −∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 18.18.18.18. 1 1 s n 2 s n 2 2 cos2 2 2 e x dx e x dx x= ⋅ = −∫ ∫ 19.19.19.19. ( ) ( ) ( ) 1 1 s n 2 6 s n 2 6 2 cos 2 6 2 2 e x dx e x dx x+ = + ⋅ = − +∫ ∫ 20.20.20.20. ( ) ( ) ( )2 2 21 1 s n 3 s n 2 cos 3 2 2 x e x dx e x x dx x⋅ + = ⋅ = − +∫ ∫ 21.21.21.21. ( ) ( ) ( )2 2 2 1 s n 1 cos 1x e x x dx x x+ ⋅ + + = − + +∫ 22.22.22.22. ( ) ( ) ( s n 1 s n cos e Lx dx e Lx dx L x x = ⋅ = −∫ ∫ 23.23.23.23. ( ) ( )s n cosx x x e e e dx e= −∫ 24.24.24.24. ( ) 1 1 c s n5 s n5 5 cos5 5 5 e x dx e x dx x= ⋅ = − =−∫ ∫ 25.25.25.25. ( ) ( ) ( ) 1 1 s n 7 8 s n 7 8 7 cos 7 8 7 7 e x dx e x dx x+ = + ⋅ =− +∫ ∫ Tipo cosenoTipo cosenoTipo cosenoTipo coseno cos s nxdx e x=∫ cos s nf f dx e f′⋅ =∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 26.26.26.26. 1 1 cos2 cos2 2 sin 2 2 2 x dx x dx x= ⋅ =∫ ∫ 27.27.27.27. ( ) ( ) ( ) 1 1 cos 2 1 cos 2 1 2 s n 2 1 2 2 x dx x dx e x+ = + ⋅ = +∫ ∫ 28.28.28.28. ( ) ( )2 21 cos 1 cos 2 x x dx x⋅ + =∫ ∫ 29.29.29.29. ( ) ( ) ( )2 2 2 1 cos 1 s n 1x x x dx e x x+ ⋅ + + = + +∫ 30.30.30.30. ( ) ( ) ( ) cos 1 cos s n Lx dx Lx dx e Lx x x = ⋅ =∫ ∫ 31.31.31.31. cos s nx x x e e dx e e=∫ 32.32.32.32. ( ) ( )2 3 3 2 3 3 cos 9 cos 9 3 s nx x dx x x dx e x+ = + ⋅ = +∫ ∫ ∫ 33.33.33.33. ( ) ( ) ( )2 3 3 2 31 1 cos 1 cos 1 3 s n 1 3 3 x x dx x xdx e x+ = + ⋅ = +∫ ∫
  6. 6. www.fisicaeingenieria.es Tabla de integrales 5 Tipo tangenteTipo tangenteTipo tangenteTipo tangente 2 sec tanxdx x=∫ 2 sec tanf f dx f′⋅ =∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 34.34.34.34. 2 2 3sec 3 sec 3tanx dx x dx x= =∫ ∫ 35.35.35.35. 2 2 2 7 7sec 7 sec 7 tan cos dx x dx x dx x x = = =∫ ∫ ∫ 36.36.36.36. ( ) ( )2 2 5 5tan 5 1 tan 5tanx dx x dx x+ = + =∫ ∫ 37.37.37.37. ( ) ( ) ( )2 2 3 2 3 2 3 9 3 sec 9 sec 9 3 tanx x dx x x dx x + + = + ⋅ =∫ ∫ 38.38.38.38. ( ) ( ) ( )2 21 1 sec 2 1 sec 2 1 2 tan 2 1 2 2 x dx x dx x+ = + ⋅ = +∫ ∫ 39.39.39.39. ( ) ( )4 2 2 2 2 2 31 sec 1 tan sec sec tan sec tan tan 3 x dx x x dx x x x x x dx= + = + = +∫ ∫ ∫ 40.40.40.40. ( )2 2 tan 1 tan 1 tanx dx x dx x x = + − = − ∫ ∫ Tipo cotangenteTipo cotangenteTipo cotangenteTipo cotangente 2 cosec cotgx dx x=−∫ 2 cosec cotgf f dx f′⋅ =−∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 41.41.41.41. 2 2 3 cosec 3 cosec 3 cotgx dx x dx x= = −∫ ∫ 42.42.42.42. 2 2 2 8 8 cosec 8 cosec 8 cotg sin dx x dx x dx x x = = = −∫ ∫ ∫ 43.43.43.43. ( ) ( )2 2 5 5 cotg 5 1 cotg 5 cotgx dx x dx x+ = + = −∫ ∫ 44.44.44.44. ( ) ( ) ( )2 2 1 cosec 2 1 cosec 2 1 2 cotg 2 1 2 x dx x dx x+ = + ⋅ = − +∫ ∫ 45.45.45.45. ( )2 2 cotg 1 cotg 1 cotgx dx x dx x x = + − = − − ∫ ∫ 46.46.46.46. ( ) ( )4 2 2 2 2 2 cosec 1 cotg cosec cosec cotg cosecx dx x x dx x x x dx= + = + =∫ ∫ ∫ 31 cotg cotg 3 x x dx− −
  7. 7. www.fisicaeingenieria.es Tabla de integrales 6 Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno) 2 1 arcsen 1 = − ∫ dx x x 2 arcsen 1 ′ = − ∫ f dx f f Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 47.47.47.47. ( ) 2 4 22 2 2 arcsen 1 1 x x dx dx x x x = = − − ∫ ∫ 48.48.48.48. ( ) 2 2 arcsen 1 1 x x x x x e e dx dx e e e = = − − ∫ ∫ 49.49.49.49. ( ) ( )2 2 1 1 arcsen 1 1 xdx dx Lx x L x Lx = = − − ∫ ∫ 50.50.50.50. ( ) 2 1 1 1 2 2arcsen 1 2 1 dx dx x x x x x = ⋅ = − − ∫ ∫ Tipo arco tangente (=Tipo arco tangente (=Tipo arco tangente (=Tipo arco tangente (=----arco cotangente)arco cotangente)arco cotangente)arco cotangente) 2 1 arctg 1 = +∫ dx x x 2 arctg 1 ′ = +∫ f dx f f EjeEjeEjeEjercicios resueltos:rcicios resueltos:rcicios resueltos:rcicios resueltos: 51.51.51.51. 2 2 1 1 1 1 arctg 3 3 3 1 3 dx dx x x x = = + +∫ ∫ 52.52.52.52. ( ) 22 1 1 3 1 arctg 3 1 9 3 31 3 dx dx x x x = = + + ∫ ∫ 53.53.53.53. ( ) ( ) 2 3 23 3 1 arctg 2 1 2 x dx x x x x + = + + + + + ∫ 54.54.54.54. ( )2 cos arctg sin 1 sin x dx x x = +∫
  8. 8. www.fisicaeingenieria.es Tabla de integrales 7 55.55.55.55. ( ) 2 24 2 1 2 1 arctg 1 2 21 x x dx dx x x x = = + + ∫ ∫ 56.56.56.56. ( ) 2 2 3 26 3 1 3 1 arctg 1 3 31 x x dx dx x x x = = + + ∫ ∫ 57.57.57.57. ( ) 22 arctg 1 1 x x x x x e e dx dx e e e = = + + ∫ ∫ INTEGRALESINTEGRALESINTEGRALESINTEGRALES INDEFINIDASINDEFINIDASINDEFINIDASINDEFINIDAS 1.1.1.1. ( ) 4 3 2 3 2 5 4 4 2 2 x x x x x x dx C+ + − = + − +∫ 2.2.2.2. 10 10 10 x x dx C ln = +∫ 3.3.3.3. ( )2 2 2 2 2 2x x x x x e dx x e x x dx x e x e dx⋅ = ⋅ − ⋅ ⋅ = ⋅ − ⋅ ⋅ = ∗∫ ∫ ∫ 2 2 x x u x du xdx dv e dx v e = ⇒ = = ⇒ = x x u x du dx dv e dx v e = ⇒ = = ⇒ = ( ) 2 2 2 2x x x x x x x e x e e dx x e x e e C   ∗ = ⋅ − ⋅ − = ⋅ − ⋅ − + =  ∫ ( )2 2 2 2 2 2x x x x x e xe e C e x x C= ⋅ − + + = − + + 4.4.4.4. ( ) ( ) 2 1 1 2 x x x e e e dx C + + = +∫ 5.5.5.5. ( ) 1 1 3 3 3 2 23 1 1 3 1 3 4 4 x x e x dx e dx x dx x dx dx dx x x x xx −− −  + − − + = + − − + =    ∫ ∫ ∫ ∫ ∫ ( ) 1 1 3 1 23 1 4 4 3 1 41 3 x x e x dx ln x x dx + −− − = + − ⋅ ⋅ − + = + ∫ ∫ ( ) 14 1 2 133 41 3 4 14 2 11 3 3 x xx x e ln x C − + − + − = + − ⋅ − + ⋅ + = − − ++ ( ) 4 2 3 3 3 3 3 4 4 8 x e x x ln x C x − − + ⋅ − − − +
  9. 9. www.fisicaeingenieria.es Tabla de integrales 8 6.6.6.6. ( ) ( ) ( ) ( ) 4 1 4 4 4 2 11 1 1 2 1 2 1 2 2 2 4 12 1 x dx x dx x dx C x − + − − + = + = + ⋅ ⋅ = ⋅ + − ++ ∫ ∫ ∫ ( ) ( ) 3 3 1 1 2 1 6 6 2 1 x C C x − = − ⋅ + + = − + + 7.7.7.7. 2 3 3 7 5 5 4 7 5 3 4 7 3 x dx ln x x C x x +   = + + +     + +    ∫ 8.8.8.8. ( ) ( ) ( ) ( ) ( ) 4 12 42 4 32 2 2 1 1 2 1 4 1 3 x xx dx x x x dx C C x x x x + ++ = + ⋅ + = + = − + − ++ + ∫ ∫ 9.9.9.9. ( )2 x dx x tgx tgx dx x tgx ln cosx C x tgx ln cosx C cos x = ⋅ − ⋅ = ⋅ − − + = ⋅ + +∫ ∫ 2 1 u x du dx v tgx dv dx cos x = ⇒ = = ⇒ = 10.10.10.10. 1 1 1 1 1 1x x x x dx dx dt e e te t e t − = = ⋅ = + + + ∫ ∫ ∫ 1 x e t x lnt dx tdt = ⇒ = = 1 x lne lnt xlne lnt lne x lnt = = ⇒ = = 2 2 1 1 1 1 1 x dt dt arctgt C arctge C t t t t ⋅ = = + = + + +∫ ∫ 11.11.11.11. ( ) ( )2 3 2 2 3 2 31 1 1 1 3 3 3 tg x x dx tg x x dx tgx C+ ⋅ ⋅ = + ⋅ ⋅ = +∫ ∫ 12.12.12.12. ( )2 1 2 1 1 2 1 2 2 2 2 2 cos x sen x sen xdx dx cos x dx x C ⋅   = = − = − +    ∫ ∫ ∫ 2 2 2 2 2 1 cos x cos x sen x cos x sen x − = − + = + 1 2 2 2 2 2 2 x senx cosx C x senx cosx C ⋅ ⋅ = − + ⋅ = − +
  10. 10. www.fisicaeingenieria.es Tabla de integrales 9 2 2 1 2 2 1 2 2 cos x sen x cos x sen x − = − = 13.13.13.13. ( ) ( )2 2 2 1 1 1tg xdx tg x dx tg x dx dx tgx x C= + − = + − = − +∫ ∫ ∫ ∫ 14.14.14.14. ( ) ( ) ( )2 2 2 2 1 1 1tg x dx tg x dx dx tg x dx x tgx C+ = + + = + + = + +∫ ∫ ∫ ∫ 15.15.15.15. 2 2 2 1 1 2 2 2 2 x x x x lnx dx lnx dx lnx xdx x ⋅ ⋅ = ⋅ − ⋅ = ⋅ − =∫ ∫ ∫ 2 2 2 2 1 2 2 2 2 4 x x x x lnx C lnx C= − ⋅ + = − + 2 1 2 u lnx du dx x x dv xdx v = ⇒ = = ⇒ = 16.16.16.16. ( ) ( ) 1 12 2 1 2 2 2 11 1 1 1 2 12 2 1 2 x x x dx x x dx C + + + ⋅ ⋅ = + ⋅ ⋅ = + = + ∫ ∫ ( ) ( ) 3 3 2 22 2 1 11 32 3 2 x x C C + + = + = + 17.17.17.17. ( ) ( ) 1 1 21 2 1 1 2 13 3 1 3 1 senxcosx dx senx cosx dx C senx − + − − +  +  = + ⋅ ⋅ = + =  +   ∫ ∫ ( ) 1 2 1 3 6 1 1 2 senx C senx C + = + = + + 18.18.18.18. ( ) 4 4 3 31 1 1 3 x x x x x x x x x x x x e e e e e dx dx e e dx x e C e e e e − − + + = + + = + + = + − +    ∫ ∫ ∫ 19.19.19.19. 2 2 2 1 1 1 13 3 3 39 9 1 9 9 3 x dx dx dx arcsen C x x x   = = = +   −  − −    ∫ ∫ ∫
  11. 11. www.fisicaeingenieria.es Tabla de integrales 10 20.20.20.20. ( ) 3 2 23 2 2 2 2 2 2 1 1 1 2 1 31 2 xx dx x x x x dx x x C x + = ⋅ + − + ⋅ ⋅ = ⋅ + − + = + ∫ ∫ 21.21.21.21. 2 2 2 2 1 1 u x du xdx x dv dx v x x = ⇒ = = ⇒ = + + 22.22.22.22. 2 2 1 1 1 1 1 1 1lnx dx lnx dx lnx dx lnx C x x x x x x x x   = − − − = − − − = − − +    ∫ ∫ ∫ 2 1 1 1 u lnx du dx x dv dx v x x = ⇒ = = ⇒ = − 23.23.23.23. ( ) ( ) ( ) ( ) 2 2 2 22 3 2 3 5 5 1 5 5 4 4 x x dx dx cos x cos xsen x sen x    + = + =     ∫ ∫ ∫ ( ) ( ) ( ) 2 3 22 3 5 3 5 4 5 5 4 3 4 4 3 4 x dx dx cotg x tg x C cos xsen x − = − + = − + +∫ ∫ 24.24.24.24. ( ) ( ) ( ) ( ) ( )2 5 2 5 2 5 2 5 2 5 1 1 1 1 2 2 2 2 x x x x x dx e dx e dx e C C e e − + − + − + + + = = − =− + = − + −∫ ∫ ∫ ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 3 2 1 1 3 2 2 1 1 3 3 x x x x C x x x C x x x x C C = ⋅ + − + ⋅ + + =   = − + + + =   − − + = + + = +
  12. 12. www.fisicaeingenieria.es Tabla de integrales 11 25.25.25.25. 2 2 22 1 1 1 1 1 1 14 22 44 4 4 2 2 1 1 4 4 2 2 x dx dx dx dx arctg C xx x x   = = = ⋅ = +  +     + + +        ∫ ∫ ∫ ∫ 26.26.26.26. 2 2 2 3 3 3 3 2 2 3 2 2 4 4 4 3 4 3 x x x dx dx dx ln x C x x x = = = + + + + +∫ ∫ ∫ 27.27.27.27. ( ) 2 2 24 2 1 1 1 1 1 1 1 2 2 1 2 21 x x x e dx dt dt arctg t C arctg e C e tt = ⋅ = = + = + + ++ ∫ ∫ ∫ 28.28.28.28. ( ) 2 2 3 3 6x x e x dx e C− − ⋅ − = +∫ 29.29.29.29. 2 2 2 1 2 tgx tg x dx tgx dx C cos x cos x = ⋅ = +∫ ∫ Otra forma de hacerla: ( ) ( ) ( ) 2 3 3 2 1 2 2 cosxsenx dx cosx senx dx C C cos x cos x − − = − − = − + = + −∫ ∫ 30.30.30.30. tgx dx ln cosx C⋅ = − +∫ 31.31.31.31. ( ) ( ) ( ) ( ) 7 3 4 7 3 4 1 1 7 7 3 4 4 7 4 1 3 7 3 4 7 4 7 4 7 4 cos x sen x dx cos x dx sen x dx cos x dx sen x dx sen x cos x sen x cos x C C − = ⋅ − ⋅ = = ⋅ ⋅ − ⋅ = ⋅ ⋅ = = − − + = + + ∫ ∫ ∫ ∫ ∫ 32.32.32.32. ( ) ( ) ( )2 22 2 2 1 2 1 4 2 21 4 1 4 x x dx dx arctg x C x x = = + + + + + + ∫ ∫ 33.33.33.33. ( ) 2 1 2 lnxlnx dx lnx dx C x x = ⋅ = +∫ ∫ 34.34.34.34. x x x x x x e cosx dx e cosx e senx dx e cosx e senx e cosx dx ⋅ ⋅ = ⋅ + ⋅ ⋅ = ⋅ + ⋅ − ⋅ ⋅  ∫ ∫ ∫ x x u cosx du senxdx dv e dx v e = ⇒ = − = ⇒ = x x u senx du cosxdx dv e dx v e = ⇒ = = ⇒ =
  13. 13. www.fisicaeingenieria.es Tabla de integrales 12 Resumiendo: 2 2 x x x x x x x x x x e cosx dx e cosx e senx e cosx dx e cosx dx e cosx e senx e cosx e senx e cosx dx C ⋅ = + − ⋅ = ⋅ = + = + ⋅ = + ∫ ∫ ∫ ∫ 35.35.35.35. ( )4 2 2 2 2 1sen x dx sen x sen x dx sen x cos x dx⋅ = ⋅ ⋅ = − =∫ ∫ ∫ ( ) 2 2 2 2 2 2 4 sen x sen x sen x cos x dx sen x dx   − ⋅ = − =    ∫ ∫ 2 2 2 2 2 2 2 1 1 2 2 1 2 2 cos x cos x sen x cos x sen x cos x sen x cos x sen x − = − + = + − = − = 1 4 1 2 2 2 4 1 2 1 4 2 2 8 8 cos x cos x dx cos x cos x dx −   − = − =        = − − + =    ∫ ∫ 2 4 3 2 4 3 2 4 8 2 8 8 2 8 sen x sen x cos x cos x dx x C   − + = − + +  −  ∫ 3 2 4 8 4 32 sen x sen x x C= − + + 36.36.36.36. senx senx e cosx dx e C⋅ ⋅ = +∫ 37.37.37.37. ( )3 3 3 2 3 2 1sen x cos x dx sen x cos x cosx dx sen x sen x cosx dx⋅ ⋅ = ⋅ ⋅ ⋅ = − ⋅ ⋅ =∫ ∫ ∫ ( ) 4 6 3 5 4 6 sen x sen x sen x cosx sen x cosx dx C= ⋅ − ⋅ = − +∫ Otra forma de hacerlo: ( )2 3 2 3 1senx sen x cos x dx senx cos x cos x dx⋅ ⋅ ⋅ = − ⋅ ⋅ =∫ ∫ ( ) ( )3 3 3 5 4 6 4 6 cos x senx dx cos x senx dx cos x senx dx cos x senx dx cos x cos x C ⋅ ⋅ − ⋅ ⋅ = − − + − = = − + + ∫ ∫ ∫ ∫ 38.38.38.38. ( ) ( ) ( ) ( )2 2 21 1 1 1 2 1 2 2 x cos x dx cos x x dx sen x C   ⋅ + = − + ⋅ − ⋅ = − + +   ∫ ∫
  14. 14. www.fisicaeingenieria.es Tabla de integrales 13 39.39.39.39. ( ) ( ) 22 1 1 1 2 2 2 2 111 dx t dt dt arctgt C arctg x C tt tx x = ⋅ ⋅ = = + = + +++ ∫ ∫ ∫ 2 2 x t x t dx tdt = ⇒ = = 40.40.40.40. 2 1 4 5 1 43 3 3 3 9 3 3 3 3 x dx dx ln x ln x C x x x  −−  = + = − − + + + − − +    ∫ ∫ 1 0 9 3 3 9 1 3 0 3 3 1 0 − − ( ) ( ) ( )( ) ( ) ( ) 2 2 5 9 3 3 3 35 9 3 3 5 3 3 13 2 6 3 43 8 6 3 x A B x x x A x B xx x x x x A x B x x A A x B B − = + − − + + + −− = − − + − = + + − = ⇒ − = ⇒ = − = − ⇒ − = − ⇒ = 41.41.41.41. 3 3 3 5 5 5 x x x x x e e dx dx ln e C e e = = + + + +∫ ∫ 42.42.42.42. ( ) ( ) ( )3 3 4 36 3 5 5 6 4 24 223 1 1 6 6 6 11 t t t tx x t t dx t dt t dt dt t t tt tx x − −− − = ⋅ = ⋅ ⋅ = = − −−− ∫ ∫ ∫ ∫ 6 6 5 6 x t t x dx t dt = ⇒ = = ( )( ) ( )( ) 4 2 6 5 4 6 3 2 6 4 3 2 4 3 26 6 6 6 6 1 1 6 6 1 1 1 1 6 1 1 3 2 3 6 6 1 2 3 2 3 6 6 1 2 t t t t t t t dt dt t t t t t t t dt t t t t t t ln t C x x x x x ln x C − + + + = = = − + +   = + − + − + =  +  = + − + − + + + = = + − + − + + + ∫ ∫ ∫
  15. 15. www.fisicaeingenieria.es Tabla de integrales 14 1 0 0 1 1 1 1 1 0 − 1 1 1 ( )( )2 2 1 1 1t t t t− = − + + 1 1 1 0 0 0 0 1 1 0 1 1 1 1 1 − − − − − 1 0 1 −1 −1 −1 43.43.43.43. ( ) 2 2 2 1 1 1 1 1 11 1 1 1 x x x x x dx dx dx dx xx x x x + + ⋅ − − − = = = = −− − ⋅ − − ∫ ∫ ∫ ∫ 2 2 1 1 x sent t arcsenx dx cost cost sen t x = ⇒ = = = − = − 2 2 1 1 1 1 sen t cos t cost cost dt cost dt cost dt sent sent sent − = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = − − −∫ ∫ ∫ ( )( )2 2 1 11 1 1 1 sent sentcos t sen t cost dt dt dt sent sent sent − +− ⋅ ⋅ = = = − − −∫ ∫ ∫ ( ) 2 1 1sent dt t cost C arcsenx x C+ = − + = − − +∫ 44.44.44.44. ( ) ( )( ) 2 2 1 11 1 1 1 1 1 1 senx senx senx dx dx dx dx senx senx senx sen x cos x ⋅ − + + = = = = − − + −∫ ∫ ∫ ∫ 2 2 2 2 1 1senx dx dx cos x senx dx cos x cos x cos x −  + = + ⋅ ⋅ =    ∫ ∫ ∫ ( ) ( ) ( ) 1 2 2 1 1 1 cosx dx cosx senx dx tgx C tgx C cos x cosx − − = − ⋅ − = − + = + + −∫ ∫ 45.45.45.45. ( ) 2 3 2 2 2 2 2 1 1 x t t t dx t dt dt dt t t t t tx x = ⋅ ⋅ = = = + + ++ ∫ ∫ ∫ ∫ 2 2 x t x t dx tdt = ⇒ = = 1 0 0 1 1 1 1 − − 1 −1 Cociente t=1
  16. 16. www.fisicaeingenieria.es Tabla de integrales 15 2 1 2 1 2 1 1 2 t t dt t ln t C t    = − + = − + + + =   +    ∫ 2 2 2 1 2 2 1t t ln t C x x ln t C= − + + + = − + + + 46.46.46.46. 2 3 2 2 2 1 3 tg x tg x dx tg x dx C cos x cos x = ⋅ ⋅ = +∫ ∫ 47.47.47.47. 2 2 1tgx tgx tgxe dx e dx e C cos x cos x = ⋅ = +∫ ∫ 48.48.48.48. 2 2 2 2 5 1 5 2 1 10 1 9 5 9 5 2 9 5 2 9 5 2 x x x dx dx dx ln x C x x x ⋅ = = = + + + + +∫ ∫ ∫ 49.49.49.49. ( ) 3 2 3 28 8 1 8 8 1 1 3 x x x x dx x dx x C x + + + = + = + + +∫ ∫ 8 8 1 1 1 8 0 1 0 − − − 8 0 1 50.50.50.50. ( )2 2 22 2 2 2 1 2sen x cos x sen xsen x sen x sen x cos x dx dx dx senx cosx senx cosx senx cosx + ++ + + = = = ⋅ ⋅ ⋅∫ ∫ ∫ 2 2 2 2 sen x cos x senx cosx dx dx dx dx senx cosx senx cosx cosx senx + = + = ⋅ ⋅∫ ∫ ∫ ∫ 2ln cosx ln senx C− + + 51.51.51.51. ( ) ( ) 4 5 5 4 1 4 4 senxcosx dx senx cosx dx C C sen x sen x − − = ⋅ ⋅ = + =− + −∫ ∫ 52.52.52.52. ( ) ( ) ( ) ( ) 1 112 22 2 1 2 2 2 1 12 1 2 1 111 2 2 x xx dx x x dx C C x − + − − − = − − − = − + = − + = − +− ∫ ∫ 2 2 1 x C= − − + 53.53.53.53. ( ) 2 4 22 2 1 2 1 1 x dx x dx arcsenx C x x = ⋅ ⋅ = + − − ∫ ∫ 54.54.54.54. 2 1 2 2 2 2 4 cos x x sen x cos x dx dx C + ⋅ = = + +∫ ∫
  17. 17. www.fisicaeingenieria.es Tabla de integrales 16 2 2 2 2 2 1 cos x cos x sen x cos x sen x = − = + 2 2 1 2 2 1 2 2 cos x cos x cos x cos x + = + = 55.55.55.55. x cosx dx xsenx senx dx xsenx cosx C⋅ ⋅ = − ⋅ = + +∫ ∫ u x du dx dv cosx dx v senx = ⇒ = = ⋅ ⇒ = 56.56.56.56. 2 2 1 1 1 arcsenx dx x arcsenx x dx x arcsenx x C x ⋅ = ⋅ − ⋅ = ⋅ + + + − ∫ ∫ 2 1 1 u arcsenx du dx x dv dx v x = ⇒ = − = ⇒ = 57.57.57.57. 3 2 2 4 4 1 1 x x dx x dx x x − −  = + =  − −  ∫ ∫ 23 5 3 52 2 1 1 1 1 2 2 2 x x dx ln x ln x C x x −   = + + = − − + + + − +    ∫ ( )( )2 1 1 1x x x− = − + 3 2 3 4 1 4 x x x x x x − − − + − 3 2 2 4 4 1 1 x x x x x − − = + − − ( ) ( ) ( )( )2 2 1 14 4 1 1 1 1 1 1 A X B Xx A B x x X X x X X + + −− − = + ⇒ = − − + − − + ( ) ( )4 1 1X A X B X− = + + − 31 3 2 2 51 5 2 2 X a A X B B = ⇒ − = ⇒ = − = − ⇒ − = − ⇒ = 58.58.58.58. 2 2 2 2 1 1 1 cos x cos xdx dx dx ln tgx C senx cosxsenx cosx tg x cos x = = = + ⋅⋅∫ ∫ ∫ 59.59.59.59. ( ) ( )2 23 1 1 1 1 senx sen x senx cos xsen x dx dx dx cosx cosx cosx − = = = − − −∫ ∫ ∫ ( )( ) ( ) 1 1 1 1 senx cosx cosx dx senx cosx dx cosx − + = + = −∫ ∫ 2 2 sen x senx dx senx cosx dx cosx C= ⋅ + ⋅ ⋅ = − + +∫ ∫
  18. 18. www.fisicaeingenieria.es Tabla de integrales 17 60.60.60.60. 2 3 2 1 2 2 2 1 1 1 11 x t t dx t dt dt t t dt t t tx   = ⋅ ⋅ = = − + − =  + + ++   ∫ ∫ ∫ ∫ 2 2x t x t dx tdt= ⇒ = ⇒ = 1 0 0 0 1 1 1 1 1 − − − 1 −1 1 − 3 2 1 1 1 1 t t t t t = − + − + + 3 2 3 2 3 2 1 3 2 2 2 2 1 3 2 2 2 1 3 t t t ln t C t t t ln t C x x x ln x C   = − + − + + =    = − + − + + = = − + − + + 61.61.61.61. ( ) ( ) 2 2 2 1 1 1 2 2 2 21 4 1 2 1 2 x x x x x x dx dx ln dx ln = ⋅ ⋅ = ⋅ ⋅ ⋅ = − − − ∫ ∫ ∫ ( )1 2 2 x arcsen C ln = ⋅ + 2 2 2 2 2 x x x dt t ln dx dt dx ln = ⇒ ⋅ = ⇒ = 62.62.62.62. ( ) ( ) ( )2 1 1 1 1 3 33x dx dt dt t t t t te − = ⋅ = = − −−∫ ∫ ∫ 1x e t x lnt dx dt t = ⇒ = ⇒ = ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 3 31 1 3 3 3 3 At t B t CtA B C t t t t t t t t t − + − + = + + ⇒ = − − − − ( ) ( ) 2 1 3 3At t B t Ct= − + − + 10 1 3 3 13 1 9 9 t B B t C C = ⇒ = − ⇒ = − = ⇒ = ⇒ = 2 1 1 1 1 1 19 3 9 3 3 9 3 9 dt ln t ln t C t t t t  − −  = + + = − + + − + = −    ∫
  19. 19. www.fisicaeingenieria.es Tabla de integrales 18 1 1 1 3 9 3 9 x x x ln e C e = − + + − + 63.63.63.63. ( ) ( ) ( ) 22 1 1 1 2 2 12 1 2 1 dx t dt dt tx x t t = ⋅ − = − = + − − − −  ∫ ∫ ∫ 2 2 1 1 1 2x t x t x t dx tdt− = ⇒ − = ⇒ = − ⇒ = − 2 2 1arctgt C arctg x C= − + = − − + 64.64.64.64. ( ) ( ) ( )224 3 2 24 1 1 4 4 4 1 1 t t t tx x t t dx t dt t dt dt t t t t tx x + ++ + = = = = − − −−∫ ∫ ∫ ∫ 4 24 4x t x t dx t dt= ⇒ = ⇒ = 1 1 0 0 0 1 2 2 2 2 Resto1 2 2 2 = 3 2 2 2 2t t t⇒ + + + 4 3 4 3 3 2 22 2 4 4 2 2 2 4 2 2 1 1 1 4 3 t t t t dt t t t dt t t ln t C t t  +   = = + + + + = + + + + − +   − −    ∫ ∫ 4 3 2 34 4 48 8 4 8 8 1 4 8 8 1 3 3 t t t t ln t C x x x x ln x C= + + + + − + = + + + + − + 65.65.65.65. ( ) 3 2 3 4 334 1 1 4 4 4 11 t t dx t dt dt dt t t tt tx x = ⋅ ⋅ = = = − −−− ∫ ∫ ∫ ∫ 4 3 4x t x t dx t dt= ⇒ = ⇒ = 2 3 34 3 4 3 4 4 1 1 3 1 3 3 t dt ln t C ln x C t = = − + = − + −∫ 66.66.66.66. 4 2 3 2 2 2 3 48 48 3 12 12 4 4 4 x dx x dx x x dx x x x   = + + = + + =  − − −  ∫ ∫ ∫
  20. 20. www.fisicaeingenieria.es Tabla de integrales 19 4 4 2 2 2 3 3 12 12 48 x x x x x − + −12 + 48 ( ) ( ) ( )( ) ( ) ( ) 2 2 48 4 2 2 2 248 4 2 2 48 2 2 2 48 4 12 2 48 4 12 A B x x x A x B x x x x A x B x x A A x B B = + − − + + + − = − − + = + + − = ⇒ = ⇒ = = − ⇒ = − ⇒ = − 3 3 92 92 12 2 2 12 12 2 12 2 x x dx x x x x ln x ln x C   = + + − =  − +  = + + − − + + ∫ 67.67.67.67. ( ) 1 1 1 1 1 2 1 1 1 1 x x e t t dx dt dt dt e t t t t t t + + +   = ⋅ = = + =  − − − −  ∫ ∫ ∫ ∫ 1x e t x lnt dx dt t = ⇒ = ⇒ = ( ) ( ) ( ) ( ) 11 1 1 1 1 1 A t Btt A B t t t t t t t t t − ++ + = + ⇒ = − − − − ( ) 0 1 1 1 1 2 t A t A t Bt t B = ⇒ = + = − + ⇒ = ⇒ = 2 1 2 1 2 1x x x ln t ln t C lne ln e C x ln e C= + − + = + − + = + − + 68.68.68.68. 1 1 1 1 2 2 2 1 1 11 1 t t dx t dt dt dt t t tx + − = ⋅ ⋅ = = = + + ++ + ∫ ∫ ∫ ∫ 1 1 1 2 2 1 1 1 t dt dt dt dt t t t +    = − = − =   + + +    ∫ ∫ ∫ ∫ 2 2 1 1 1 2x t x t x t dx tdt+ = ⇒ + = ⇒ = − ⇒ = 69.69.69.69. ( )2 2 2 1 3 1 3 1 3 3 1 1 1 x dx dx ln x ln x C x x x x x x + − −  = + + = − + + − +  − −  ∫ ∫ ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 1 12 1 2 1 1 1 1 1 Ax x B x Cxx A B C x x x x x x x x x x − + − ++ + = + + ⇒ = = − − − −
  21. 21. www.fisicaeingenieria.es Tabla de integrales 20 ( ) ( ) 2 2 1 1 1 0 1 1 1 3 3 2 5 2 4 5 2 1 12 6 2 3 x Ax x B x Cx x B B x C C x A B C A A A + = − + − + = ⇒ = − ⇒ = − = ⇒ = ⇒ = = ⇒ = + + ⇒ = − + ⇒ − = ⇒ = − 70.70.70.70. 2 2 1 1 1 2 2 2 1 1 1 dx t dt dt arcsent C x x t t t = ⋅ ⋅ = = + = − − − ∫ ∫ ∫ 2arcsen x C= + 2 2x t x t dx tdt= ⇒ = ⇒ = 71.71.71.71. ( )3 2 2 1cos x dx cos x cosx dx sen x cosx dx⋅ = ⋅ ⋅ = − ⋅ ⋅ =∫ ∫ ∫ ( ) 3 2 3 sen x cosx sen x cosx dx senx C= − ⋅ = − +∫ 72.72.72.72. 3 3 3 3 33 3 3 3 3 sen x xcos x cos x xcos x x sen x dx dx C − ⋅ ⋅ = + = − + + =∫ ∫ 3 3 3 u x du dx cos x dv sen x dx v = ⇒ = = ⋅ ⇒ = − 3 3 3 9 xcos x sen x C= − + + 73.73.73.73. 2 2 1 1 2 1 2 1 x arctgx dx x arctgx x x x arctgx dx x x ⋅ = ⋅ − ⋅ = ⋅ − = + +∫ ∫ ∫ 2 1 1 u arctgx du dx x dv dx v x = ⇒ = + = ⇒ = 21 1 2 x arctgx ln x C= ⋅ − + + 74.74.74.74. 2 2 2 1 1x dx sen t cost dt cos t cost dt cost cost dt− = − ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ =∫ ∫ ∫ ∫ 2 2 1 1 x sent t arcsenx dx costdt cost sen t x = ⇒ = = ⇒ = − = − 2 2 2 2 2 1 cos t cos t sen t cos t sen t = − = + 2 2 1 2 2 1 2 2 cos t cos t cos t cos t + = + =
  22. 22. www.fisicaeingenieria.es Tabla de integrales 21 2 2 2 1 2 2 2 2 2 2 2 2 sen t sent cost cos t t t cos t dt dt C C ⋅ + = ⋅ = = + + = + + =∫ ∫ 2 1 2 2 2 2 t sent cost arcsenx x x C C ⋅ ⋅ − = + + = + + 75.75.75.75. ( )2 2 2 2 2 2 2 1a x dx a a sen t a cost dt a sen t a cost dt− = − ⋅ ⋅ ⋅ = − ⋅ ⋅ ⋅ =∫ ∫ ∫ x x a sent t arcsen dx a cost dt a = ⋅ ⇒ = ⇒ = ⋅ ⋅ 2 2 a cos t a cost dt a cost a cost dt⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =∫ ∫ 2 2 2 21 2 2 2 2 4 cos t t sen t a cos t dt a dt a C +   = ⋅ = = + +    ∫ ∫ 2 2 2 1 2 2 2 2 x xx arcsen a at sent cost aa C a C    −  ⋅     = + + = + + =           2 2 2 2 2 x a arcsen x a xa C ⋅ ⋅ − = + + 76.76.76.76. ( ) ( ) 2 3 2 2 2 2 2 3 x x x x xe x e x dx e x xe dx x e dx+ = + + = + + ⋅ ⋅ =∫ ∫ ∫ x x u x du dx dv e dx v e = ⇒ = = ⇒ = ( ) 2 3 2 3 2 2 2 2 3 2 3 x x x x x xe x e x x e e dx xe e C= + + ⋅ − = + + − +∫ 77.77.77.77. ( )4 2 2 2 2 1sen x dx sen x sen x dx sen x cos x dx⋅ = ⋅ ⋅ = − =∫ ∫ ∫ ( )2 2 2 2 2 4 sen x sen x sen xcos x dx sen x dx   − = − =    ∫ ∫ 2 2 2 2 2 2 1 1 2 2 cos x cos x sen x cos x sen x cos x sen x − = + = + − = 21 2 2 cos x sen x − =
  23. 23. www.fisicaeingenieria.es Tabla de integrales 22 1 2 1 4 2 4 2 8 2 4 8 32 cos x cos x x sen x x sen x dx C − −  = − = − − + +    ∫ 3 2 4 8 4 32 x sen x sen x C= − + + 78.78.78.78. 2 2 2 2 2ln x dx x ln x lnx dx xln x x lnx dx ⋅ = ⋅ − ⋅ = − ⋅ − =  ∫ ∫ ∫ 2 1 2u ln x du lnx dx x dv dx v x = ⇒ = ⋅ ⋅ = ⇒ = 1 u lnx du dx x dv dx v x = ⇒ = = ⇒ = 2 2 2x ln x lnx x C= ⋅ − + + 79.79.79.79. 3 3 2 3x senx dx x cosx x cosx dx⋅ ⋅ = − ⋅ + ⋅ ⋅ =∫ ∫ 3 2 3u x du x dx dv senx dx v cosx = ⇒ = = ⋅ ⇒ = − 2 2u x du xdx dv cosx dx v senx = ⇒ = = ⋅ ⇒ = u x dv dx du senxdx v cosx = ⇒ = = ⇒ = − 3 2 3 2x cosx x senx x senx dx = − ⋅ + − ⋅ ⋅ = ∫ 3 2 3 6x cosx x senx x cosx cosx dx − ⋅ + ⋅ − − ⋅ + ⋅ =  ∫ 3 2 3 6 6x cosx x senx x cosx senx C= − ⋅ + ⋅ + ⋅ − + 80.80.80.80. 2 2 2 1 2 2 2 2 2 2 2 x x x x x x x x dx dx C ln ln ln ln ln − − − − − ⋅ ⋅ − ⋅ = − + = − + ⋅ + =∫ ∫ 2 2 2 x x u x du dx dv dx v ln − − = ⇒ = = ⋅ ⇒ = − ( ) 2 2 2 2 2 x x x C ln ln − − ⋅ = − + 81.81.81.81. 3 3 3 3 3 3 3 3 3x x x x x x cosx dx senx ln senx dx senx ln cosx ln cosx d⋅ ⋅ = ⋅ − ⋅ ⋅ = ⋅ − − ⋅ + ⋅ ⋅ ∫ ∫ ∫ 3 3 3x x u du ln dx dv cosx dx v senx = ⇒ = ⋅ ⋅ = ⋅ ⇒ = 3 3 3x x u du ln dx dv senx dx v cosx = ⇒ = ⋅ = ⋅ ⇒ = − ( ) 2 3 3 3 3 3 3x x x x cosx dx senx ln cosx ln cosx dx⋅ = + ⋅ ⋅ − ⋅ ⋅ =∫ ∫ ( ) 2 1 3 3 3 3 3x x x ln cosx dx senx ln cosx + ⋅ ⋅ = + ⋅ =  ∫
  24. 24. www.fisicaeingenieria.es Tabla de integrales 23 ( ) 2 3 3 3 3 1 3 x x x senx ln cosx cosx dx C ln + ⋅ ⋅ ⋅ = + + ∫ 82.82.82.82. 3 3 3 2 21 1 3 3 3 3 x x x x lnx dx lnx dx lnx x dx x ⋅ = ⋅ − ⋅ ⋅ = − =∫ ∫ ∫ 3 2 1 3 u lnx du dx x x dv x dx v = ⇒ = = ⇒ = 3 3 3 3 1 3 3 3 3 9 x x x x lnx C lnx C= − ⋅ + = − + 83.83.83.83. 2 2 1 1 1 2 2 2 dx dx tg x C x cos x cos x x = ⋅ ⋅ = + ⋅ ∫ ∫ 84.84.84.84. ( )2 2 1 1 1 1 1 dx dx arcsen lnx C xx ln x ln x = ⋅ ⋅ = + − − ∫ ∫ 85.85.85.85. ( ) ( ) 2 2 1 7 1 1 27 7 7 2 49 7 72 7 1 2 x x x x x dx dx ln dx ln x = ⋅ ⋅ = ⋅ ⋅ ⋅ = + + + ∫ ∫ ∫ 2 2 1 1 1 1 1 1 7 7 7 7 2 7 2 7 2 27 7 1 1 2 2 x x x x ln ln dx dx ln ln ⋅ = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =     + +        ∫ ∫ 2 7 2 2 7 2 7 2 7 22 x x arctg C arctg C ln ln    ⋅ = ⋅ + = ⋅ +       86.86.86.86. 2 3 2 1 1 2 5 1 2 1 12 2 2 1 2 2 1 2 2 2 x x dx dx ln x ln x ln x C x x x x x x   − + − = + + = + − − + +  + − − +    ∫ ∫
  25. 25. www.fisicaeingenieria.es Tabla de integrales 24 1 1 2 0 0 0 0 0 1 1 2 0 1 1 2 1 2 0 2 2 0 − − − − 1 ( )( ) ( )( ) ( ) ( ) ( )( ) 3 2 2 3 2 2 3 2 2 1 2 2 5 1 2 1 2 1 2 2 12 5 1 2 1 2 x x x x x x x x A B C x x x x x x A x x Bx x Cx xx x x x x x x x + − = − + + − = + + + − − + − + + + + −+ − = + − − + ( )( ) ( ) ( )2 2 5 1 1 2 2 1x x A x x Bx x Cx x+ − = − + + + + − 10 1 2 2 1 6 3 2 12 3 6 2 x A A x B B x C C = ⇒ − = − ⇒ = = ⇒ = ⇒ = = − ⇒ − = ⇒ = − 87.87.87.87. 2 2 1 1 1 1 1 1 1 1 x x e t t dx dt dt dt ln t C e t t t t −  = ⋅ = = + = − + +  + + + +  ∫ ∫ ∫ ∫ 1x e t x lnt dx dt t = ⇒ = ⇒ = 1x x e ln e C= − + + 1 0 1 1 1 − − 1 − 88.88.88.88. ( ) ( ) 5 3 2 4 2 1 1 2 2 2 5 3 t t x x dx t t tdt t t dt C   − ⋅ = + ⋅ = + = + + =    ∫ ∫ ∫ 2 2 1 1 1 2x t x t x t dx tdt− = ⇒ − = ⇒ = + ⇒ = ( ) ( ) 5 3 5 3 2 1 2 12 2 5 3 5 3 x xt t C C − − = + + = + +
  26. 26. www.fisicaeingenieria.es Tabla de integrales 25 89.89.89.89. 2 2 2 2 2 2 2 2 2 2 a sen t a a a b x dx a b cost dt a a sen t cost dt b b b − ⋅ = − ⋅ ⋅ ⋅ = − ⋅ ⋅ ⋅ =∫ ∫ ∫ 2 2 2 2 2 2 1 asent a x dx costdt b b bx b x a b x sent cost a a a bx t arcsen a = ⇒ = − = ⇒ = − =   =     ( )2 2 2 2 1 a a a sen t cost dt a cos t cost dt b b = − ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ =∫ ∫ 2 2 2 1 2 2 a a a cos t a cost cost dt cos t dt dt b b b + = ⋅ ⋅ ⋅ ⋅ = ⋅ = =∫ ∫ ∫ 2 2 2 2 2 2 4 2 2 a t a sen t a a sent cost C t C b b b b ⋅ = ⋅ + ⋅ + = ⋅ + ⋅ + = 2 2 2 2 2 2 2 a b a bx a b x arcsen x C b a b a a −  = + ⋅ ⋅ + =    2 2 2 2 2 2 b a arcsen x x a b xa C b   ⋅   − = + + 90.90.90.90. 2 1 1 x x x x e dx e ln e C e = − + + +∫ (Es la misma que la nº 87) 91.91.91.91. ( ) 4 3 2 4 3 2 3 2 2 3 2 3 1 2 3 7 7 7 4 3 2 4 3 2 x x x x x x x x x lnx dx x lnx x x     − + − ⋅ = − + − − − + −        ∫ ∫ ( ) 4 3 2 3 2 1 2 3 2 3 7 7 4 3 2 u lnx du dx x x x x dv x x x dx v x = ⇒ = = − + − ⇒ = − + − 4 3 2 3 2 2 3 2 3 7 7 4 3 2 4 3 2 x x x x x x x lnx dx     = − + − − − + − =        ∫ 4 3 2 4 3 2 2 3 2 3 7 7 4 3 2 16 9 4 x x x x x x x lnx x C     = − + − − − + − +        92.92.92.92. ( )1 1 1 1 3 3 3 2 2 4 5 5 4 7 7 x dx x x dx x − −  − = ⋅ − ⋅ ⋅ =    ∫ ∫
  27. 27. www.fisicaeingenieria.es Tabla de integrales 26 1 41 11 1 3 32 21 11 1 3 32 2 5 4 7 5 4 7 1 1 4 11 1 3 2 3 2 x x x x C C + − + − − = ⋅ − ⋅ ⋅ + = ⋅ − ⋅ ⋅ + = + − + 433 33 8 3 5 5 8 4 4 77 x x x C x x C− ⋅ + = − + = 3 33 8 7 3 8 5 5 7 4 7 4 7 x x x C x x x C− ⋅ + = − +

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