`int tan^5(x) sec^3(x) dx` Evaluate the integral

`inttan^5(x)sec^3(x)dx`

Rewrite the integrand as,

`=inttan^4(x)tanx(x)sec^3(x)dx`

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

`=int(sec^2(x)-1)^2sec^3(x)tan(x)dx`

Now apply the integral substitution,

Let `u=sec(x)`

`du=sec(x)tan(x)dx`

`=int(u^2-1)^2u^2du`

`=int(u^4-2u^2+1)u^2u`

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

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

`=u^7/7-2(u^5/5)+u^3/3`

Substitute back `u=sec(x)`

`=1/7sec^7(x)-2/5sec^5(x)+1/3sec^3(x)`

Add a constant C to the solution,

`=1/7sec^7(x)-2/5sec^5(x)+1/3sec^3(x)+C`