Question Number 136929 by liberty last updated on 27/Mar/21
$$\int\:\frac{\mathrm{e}^{\mathrm{ln}\:\left(\mathrm{sin}^{−\mathrm{1}} \mathrm{x}\right)} }{\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:=\:?\: \\ $$
Answered by Dwaipayan Shikari last updated on 27/Mar/21
$$\int\frac{{e}^{{log}\left({sin}^{−\mathrm{1}} {x}\right)} }{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx}\:\:\:\:\:{sin}^{−\mathrm{1}} {x}={t} \\ $$$$\int{e}^{{log}\left({t}\right)} {dt}=\frac{{t}^{\mathrm{2}} }{\mathrm{2}}+{C}=\frac{\left({sin}^{−\mathrm{1}} {x}\right)^{\mathrm{2}} }{\mathrm{2}}+{C} \\ $$
Answered by liberty last updated on 27/Mar/21
$$\int\:\frac{\mathrm{e}^{\mathrm{ln}\:\left(\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{x}\right)\right)} }{\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:=\:\int\:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{x}\right)\:\mathrm{d}\left(\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{x}\right)\right) \\ $$$$=\:\mathrm{2}\left(\frac{\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{x}\right)}{\mathrm{2}}\right)^{\mathrm{2}} +\:\mathrm{C} \\ $$