∫(tanx)^5.(secx)^3 dx=∫(tanx)^4.(secx)^2 d(secx)=∫[(secx)^2-1]^2.(secx)^2 d(secx)=∫[(secx)^6-2(secx)^4+(secx)^2] d(secx)=(1/7)(secx)^7-(2/5)(secx)^4 +(1/3)(secx)^3 + C