Evaluate the indefinite integral

Question

∫tan^3(x)*sec^3(x)dx?

Answer 1

The given equation is ∫tan^3(x)*sec^3(x) dx

                               =>    ∫tan^2(x)*sec^2(x)*(tanx*secx) dx          [Split tan^3(x) = tan^2(x)*tan(x) and tan^3(x) = tan^2(x)*tan(x) ]

                               =>    ∫(sec^2(x) - 1)*sec^2(x)*(tanx*secx) dx   [ Since  tan^2(x) = sec^2(x) - 1]

                               =>    ∫(sec^4(x) - sec^2(x))*(tanx*secx) dx     

                               =>    ∫(sec^4(x)*(tanx*secx) dx - ∫sec^2(x))*(tanx*secx) dx

                               =>    ∫(sec^4(x)*d(secx) - ∫sec^2(x))d(secx)     [ Since d/dx(secx) = tanx*secx ]

                              =>    (1/5)sec^5(x) - (1/3)sec^3(x) + c