integration help!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Question

∫sec^3x tan^3x dx

Answer 1

Given that ∫tan^3(x)*sec^3(x) dx

               = ∫tan^2(x)*sec^2(x)*(tanx*secx) dx

               = ∫(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 (tanx*secx) dx = d(secx) ]

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

Therefore ∫tan^3(x)*sec^3(x) dx = (1/5)sec^5(x) - (1/3)sec^3(x) + c