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Question Number 136929 by liberty last updated on 27/Mar/21

$$\int \frac{{\mathrm{e}}^{\ln \left({\sin }^{-1}\mathrm{x}\right)}}{\sqrt{1-{\mathrm{x}}^2}}\mathrm{dx}=?$$

Answered by Dwaipayan Shikari last updated on 27/Mar/21

$$\int \frac{e^{log\left({sin}^{-1}x\right)}}{\sqrt{1-x^2}}dx{sin}^{-1}x=t$$$$\int e^{log\left(t\right)}dt=\frac{t^2}{2}+C=\frac{{\left({sin}^{-1}x\right)}^2}{2}+C$$

Answered by liberty last updated on 27/Mar/21

$$\int \frac{{\mathrm{e}}^{\ln \left({\sin }^{-1}\left(\mathrm{x}\right)\right)}}{\sqrt{1-{\mathrm{x}}^2}}\mathrm{dx}=\int {\sin }^{-1}\left(\mathrm{x}\right)\mathrm{d}\left({\sin }^{-1}\left(\mathrm{x}\right)\right)$$$$=2{\left(\frac{{\sin }^{-1}\left(\mathrm{x}\right)}{2}\right)}^2+\mathrm{C}$$

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