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 x2 dx =12a

ln

x ax + a+ c

19.

1

x2 a2 dx =1

2aln

x ax + 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)]