Universidad Nacional Andres BelloDepartamento de Matematicas

Calculo IntegralProfesor Javier Olivos

Resumen de Integrales. Autor: Mauricio Vargas

Integrales Basicas

1.

∫dx = x + c

2.

∫kdx = kx + c (k cte)

3.

∫xndx =

xn+1

n + 1+ c, n 6= −1

4.

∫1

xdx = ln(|x|) + c

5.

∫eaxdx =

eax

a+ c

6.

∫abxdx =

abx

ln(a) · b+ c, a > 0

7.

∫sen(x)dx = − cos(x)dx + c

8.

∫cos(x)dx = sen(x) + c

9.

∫tan(x)dx = ln | sec(x)|+ c

10.

∫cotan(x)dx = ln | sen(x)|+ c

11.

∫sec(x)dx = ln | sec(x) + tan(x)|+ c

12.

∫cosec(x) = ln | cosec(x)− cotan(x)|+ c

13.

∫sec2(x)dx = tan(x) + c

14.

∫cosec2(x)dx = −cotan(x) + c

15.

∫sec(x) tan(x)dx = sec(x) + c

16.

∫cosec(x)cotan(x)dx = − cosec(x) + c

17.

∫1

a2 + x2dx =

1

aarctan

(xa

)+ c

18.

∫1

a2 − x2dx =

−1

2aln

∣∣∣∣x− a

x + a

∣∣∣∣+ c

19.

∫1

x2 − a2dx =

1

2aln

∣∣∣∣x− a

x + a

∣∣∣∣+ c

Sustitucion ∫g(f(x)) · f ′(x)dx =

∫g(u)du el cambio de variable es u = f(x)

Integracion por partes ∫udv = uv −

∫vdu

Identidades trigonometricas

1. sen2(x) + cos2(x) = 1

2. tan(x) =sen(x)

cos(x)

3. cotan(x) =cos(x)

sen(x)

4. sec(x) =1

cos(x)

5. cosec(x) =1

sec(x)

6. 1 + tan2(x) = sec2(x)7. 1 + cotan2(x) = cosec2(x)8. sen(2x) = 2 sen(x) cos(x)9. cos(2x) = cos2(x)− sen2(x)

10. tan(2x) =2 tan(x)

1− tan2(x)

11. sen2(x) =1− cos(2x)

2

12. cos2(x) =1 + cos(2x)

2

13. tan2(x) =1− cos(2x)

1 + cos(2x)

14. sen(2x) =2 tan(x)

1 + tan2(x)

15. cos(2x) =1− tan2(x)

1 + tan2(x)

Identidades adicionales

1. sen(x± y) = sen(x) cos(y)± sen(y) cos(x)2. cos(x± y) = cos(x) cos(y)∓ sen(y) sen(x)

3. tan(x± y) =tan(x)± tan(y)

1∓ tan(x) tan(y)

4. sen(x) sen(y) =1

2[cos(x− y)− cos(x + y)]

5. sen(x) cos(y) =1

2[sen(x + y) + sen(x− y)]

6. cos(x) cos(y) =1

2[cos(x + y) + cos(x− y)]