Question

How do you integrate `(tanx)^8`?

Answer 1

`tan^7x/7-tan^5x/5+tan^3x/3-tanx+x+C.`

Explanation:

Let, `I=inttan^8xdx=inttan^6x(sec^2x-1)d,`

`=int(tanx)^6d(tanx)-inttan^4x(sec^2x-1)dx,`

`=tan^7x/7-int(tanx)^4d(tanx)+inttan^2x(sec^2x-1)dx,`

`=tan^7x/7-tan^5x/5+int(tanx)^2d(tanx)-inttan^2xdx,`

`=tan^7x/7-tan^5x/5+tan^3x/3-int(sec^2x-1)dx.`

`:. I=tan^7x/7-tan^5x/5+tan^3x/3-tanx+x+C.`

Enjoy Maths.!