Question

How do you integrate `(tanx)^5*(secx)^4dx`?

Answer 1

`tan^8x/8+tan^6x/6+C`

Explanation:

Writing this as:

`inttan^5xsec^4xdx`

Rewriting to leave all `tanx` functions with one cluster of `sec^2x`, which is the derivative of `tanx`:

`=inttan^5xsec^2xsec^2xdx`

`=inttan^5x(tan^2x+1)sec^2xdx`

Letting `u=tanx` so `du=sec^2xdx`:

`=intu^5(u^2+1)du`

`=intu^7du+intu^5du`

`=u^8/8+u^6/6+C`

`=tan^8x/8+tan^6x/6+C`