Question Number 215687 by Ajeemkhan last updated on 15/Jan/25
Answered by som(math1967) last updated on 15/Jan/25
$$\int{sec}^{{p}−\mathrm{1}} {xsecxtanxdx} \\ $$$$\Rightarrow\int{sec}^{{p}−\mathrm{1}} {xd}\left({secx}\right) \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{{p}}{sec}^{{p}} {x}+{C} \\ $$
Commented by Ajeemkhan last updated on 15/Jan/25
$$\mathrm{Answer}\:\mathrm{is}\:\mathrm{right}\:\mathrm{but}\:\mathrm{method}\:\mathrm{not}\:\mathrm{understood} \\ $$
Commented by Ajeemkhan last updated on 15/Jan/25
$$\mathrm{Answer}\:\mathrm{is}\:\mathrm{right}\:\mathrm{but}\:\mathrm{method}\:\mathrm{not}\:\mathrm{understood} \\ $$
Commented by som(math1967) last updated on 15/Jan/25
$$\:\int\mathrm{sec}\:^{{p}−\mathrm{1}} {xsecxtanxdx} \\ $$$$\:{let}\:{secx}={t}\Rightarrow{secxtanxdx}={dt} \\ $$$$\therefore\:\int{t}^{{p}−\mathrm{1}} {dt}=\frac{\mathrm{1}}{{p}}{t}^{{p}} +{C}=\frac{\mathrm{1}}{{p}}\mathrm{sec}\:^{{p}} {x}+{C} \\ $$
Answered by MATHEMATICSAM last updated on 15/Jan/25
$$\mathrm{Put}\:\mathrm{sec}{x}\:=\:{t}. \\ $$$$\Rightarrow\:\mathrm{sec}{x}\mathrm{tan}{x}\:{dx}\:=\:{dt} \\ $$$$ \\ $$$$\int\:\mathrm{sec}^{{p}} {x}\mathrm{tan}{x}\:{dx} \\ $$$$=\:\int\:\mathrm{sec}^{{p}\:−\:\mathrm{1}} {x}.\:\mathrm{sec}{x}\mathrm{tan}{x}\:{dx} \\ $$$$=\:\int\:{t}^{{p}\:\:−\:\mathrm{1}} \:{dt} \\ $$$$=\:\frac{{t}^{{p}} }{{p}}\:+\:\mathrm{C} \\ $$$$=\:\frac{\mathrm{sec}^{{p}} {x}}{{p}}\:+\:\mathrm{C} \\ $$