`int tan^4(x) sec^6(x) dx` Evaluate the integral

`inttan^4(x)sec^6(x)dx`

Rewrite the integrand as,

`=inttan^4(x)sec^2(x)sec^4(x)dx`

use the identity: `sec^2(x)=1+tan^2(x)`

`=inttan^4(x)sec^2(x)(1+tan^2(x))^2dx`

Now apply integral substitution,

Let `u=tan(x)`

`=>du=sec^2(x)dx`

`=intu^4(1+u^2)^2du`

`=intu^4(1+2u^2+u^4)du`

`=int(u^4+2u^6+u^8)du`

`=intu^4du+2intu^6du+intu^8du`

`=u^5/5+2(u^7/7)+u^9/9`

Substitute back `u=tan(x)`

`=1/5tan^5(x)+2/7tan^7(x)+1/9tan^9(x)`

Add a constant C to the solution,

`=1/5tan^5(x)+2/7tan^7(x)+1/9tan^9(x)+C`