## Integration of Trigonometric functions-Integration Formulas

### 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:

$$\int (2+ \tan x)^2 \ dx = \int (4+4 \tan x + \tan^2x) \ dx$$
$$= 4 \int 1 \ dx + 4 \int \tan x \ dx + \int \tan^2x \ dx$$
$$= 4x+4 \int \tan x \ dx + \int \tan^2x \ dx$$
$$= 4x+4 \ln \vert \sec x \vert + \int (sec^2x-1) \ dx$$
$$= 4x+4 \ln \vert \sec x \vert + \int sec^2x \ dx – \int 1 \ dx$$
$$= 4x+4 \ln \vert \sec x \vert + \int sec^2x \ dx – x$$
$$= 3x+4 \ln \vert \sec x \vert + \int sec^2x \ dx$$
$$= 3x+4 \ln \vert \sec x \vert + \tan x + C$$