Integration of Trigonometric functions
$$ 1. \ \int \cos x \ dx = \sin x+C $$
$$ 2. \ \int \sin x \ dx=- \cos x +C $$
$$ 3. \ \int \sec^2x \ dx= \tan x +C $$
$$ 4. \ \int cosec^2x \ dx= – \cot x +C $$
$$ 5. \ \int \sec x \tan x \ dx= \sec x +C $$
$$ 6. \ \int cosec \ x \cot x \ dx= -cosec \ x +C $$
$$ 7. \ \int \tan x \ dx= \ln \vert \sec x \vert +C $$
$$ 8. \ \int \cot x \ dx= \ln \vert \sin x \vert +C $$
$$ 9. \ \int \sec x \ dx= \ln \vert \sec x + \tan x \vert +C $$
$$ 10. \ \int cosec \ x \ dx = \ln \vert cosec \ x – \cot x \vert +C $$
$$ 11. \ \int \sin^2x \ dx = \frac x2 – \frac 14 \sin 2x +C $$
$$ 12. \ \int \cos^2x \ dx = \frac x2 + \frac 14 \sin 2x +C $$
$$ 13. \ \int \sin^3x \ dx = \frac {1}{12} \cos 3x – \frac 34 \cos x +C $$
$$ 14. \ \int \cos^3x \ dx = \frac {1}{12} \sin 3x + \frac 34 \sin x +C $$
$$ 15. \ \int \sec^3x \ dx = \frac { \sin x}{2 cos^2x}+ \frac 12 \ln \left\vert \tan \left( \frac x2 + \frac {\pi}{4} \right) \right\vert +C $$
$$ 16. \ \int cosec^3x \ dx = – \frac { \cos x}{2 sin^2x}+ \frac 12 \ln \left\vert \tan \frac x2 \right\vert +C $$
$$ 17. \ \int \sin x \cos x \ dx= – \frac 14 \cos 2x +C $$
$$ 18. \ \int \sin^2x \cos x \ dx= \frac 13 \sin^3x +C $$
$$ 19. \ \int \cos^2x \sin x \ dx= – \frac 13 \cos^3x +C $$
$$ 20. \ \int \sin^2x \cos^2x \ dx= \frac x8 – \frac {1}{32} \sin 4x +C $$
$$ 21. \ \int \frac {\sin x} {\cos^2x} \ dx= \sec x +C $$
$$ 22. \ \int \frac {\sin^2x} {\cos x} \ dx= \ln \left\vert \tan \left( \frac x2 + \frac {\pi}{4} \right) \right\vert – \sin x +C $$
$$ 23. \ \int \tan^2x dx= \tan x-x +C $$
$$ 24. \ \int \frac {\cos x} {\sin^2x} \ dx = -cosec \ x +C $$
$$ 25. \ \int \frac {\cos^2x} {\sin x} \ dx = \ln \left\vert \tan \frac x2 \right\vert + \cos x +C $$
$$ 26. \ \int \cot^2 x \ dx = – \cot x -x +C $$
$$ 27. \ \int \frac {dx}{ \cos x \sin x} = \ln \left\vert \tan x \right\vert +C $$
$$ 28. \ \int \frac {dx}{ \sin^2x \cos x } = – \frac {1}{ \sin x} + \ln \left\vert \tan \left( \frac x2 + \frac {\pi}{4} \right) \right\vert +C $$
$$ 29. \ \int \frac {dx}{ \sin x \cos^2x } = \frac {1}{ \cos x} + \ln \left\vert \tan \frac x2 \right\vert +C $$
$$ 30. \ \int \frac {dx}{ \sin^2x \cos^2x }= \tan x – \cot x +C $$
$$ 31. \ \int \sin mx \sin nx \ dx= – \frac { \sin (m+n)x}{2(m+n)} + \frac {\sin (m-n)x}{2(m-n)} +C, \ m^2 \neq n^2 $$
$$ 32. \ \int \sin mx \cos nx \ dx= – \frac { \cos (m+n)x}{2(m+n)} – \frac {\cos (m-n)x}{2(m-n)} +C, \ m^2 \neq n^2 $$
$$ 33. \ \int \cos mx \cos nx \ dx= \frac { \sin (m+n)x}{2(m+n)} + \frac {\sin (m-n)x}{2(m-n)} +C, \ m^2 \neq n^2 $$
$$ 34. \ \int \sin x \cos^nx \ dx=- \frac {cos^{n+1}x}{n+1} +C $$
$$ 35. \ \int \sin^nx \cos x \ dx= \frac {sin^{n+1}x}{n+1} +C $$
Example 1:
$$ Integrate \int (3 \sin x-4 \sec^2x) \ dx $$
Solution:
$$ \int (3 \sin x-4 \sec^2x) \ dx = \int 3 \sin x \ dx – \int 4 \sec^2x \ dx $$
$$ = 3 \int \sin x \ dx – 4 \int \sec^2x \ dx $$
$$ = -3 \cos x – 4 \tan x +C $$
Example 2:
$$ Integrate \int (2+ \tan x)^2 \ dx $$
Solution:
$$ \int (2+ \tan x)^2 \ dx = \int (4+4 \tan x + \tan^2x) \ dx $$
$$ = 4 \int 1 \ dx + 4 \int \tan x \ dx + \int \tan^2x \ dx $$
$$ = 4x+4 \int \tan x \ dx + \int \tan^2x \ dx $$
$$ = 4x+4 \ln \vert \sec x \vert + \int (sec^2x-1) \ dx $$
$$ = 4x+4 \ln \vert \sec x \vert + \int sec^2x \ dx – \int 1 \ dx $$
$$ = 4x+4 \ln \vert \sec x \vert + \int sec^2x \ dx – x $$
$$ = 3x+4 \ln \vert \sec x \vert + \int sec^2x \ dx $$
$$ = 3x+4 \ln \vert \sec x \vert + \tan x + C $$