Question Number 495 by 13/NaSaNa(N)056565 last updated on 25/Jan/15
$$\int\mathrm{sec}^{\mathrm{3}} {xdx} \\ $$
Answered by prakash jain last updated on 15/Jan/15
$$\mathrm{tan}\:{x}={t} \\ $$$$\mathrm{sec}^{\mathrm{2}} {x}\:{dx}={dt} \\ $$$$\mathrm{sec}\:{x}=\sqrt{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$$\int\mathrm{sec}^{\mathrm{3}} {x}\:{dx}=\int\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$=\frac{{t}}{\mathrm{2}}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\mid{t}+\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\mid+{C} \\ $$$$=\frac{\mathrm{tan}\:{x}\:\mathrm{sec}\:{x}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\mid\mathrm{sec}\:{x}+\mathrm{tan}\:{x}\mid+{C} \\ $$