What is the Integral of ##tan^2 x sec^4 x dx##?

##tan^3x/3 + tan^5x/5 + C## where C = constant of integration

Using the Pythagorean identity, ##sec^2x = 1 + tan^2x## we have ##int tan^2 x sec^4x dx = int tan^2x sec^2 x sec^2x dx = inttan^2x(1+tan^2x)sec^2xdx## Now let ##tanx = u, ##so ##(du)/dx = sec^2x or du = sec^2x dx## Now integrating with respect to u, we have ##intu^2(1+u^2)du == int (u^2 + u^4) du = u^3/3 + u^5/5 + C ##

where C = constant of integration.

Resubstituting u = tanx, we have the final integration answer as

##tan^3x/3 +tan^5x/5 + C##