Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 163886 by zakirullah last updated on 11/Jan/22

$$Evaluatethefollowingbyusing$$$$integrationbypartsformula:$$$$\int {xsin}^{-1}\left(x\right)dx$$

Commented by Ar Brandon last updated on 11/Jan/22

$$u\left(x\right)={\sin }^{-1}\left(x\right),v^{\prime }\left(x\right)=x$$

Commented by zakirullah last updated on 11/Jan/22

$$dear\mathit{S}ir!needsolution?$$

Answered by Ar Brandon last updated on 11/Jan/22

$$\int x{\sin }^{-1}\left(x\right)dx=\frac{1}{2}{x}^2{\sin }^{-1}\left(x\right)-\frac{1}{2}\int \frac{x^2}{\sqrt{1-x^2}}dx$$$$=\frac{1}{2}{x}^2{\sin }^{-1}\left(x\right)-\frac{1}{2}(x\sqrt{1-x^2}-\int \sqrt{1-x^2}dx)$$$$\int \sqrt{1-x^2}dx=\frac{1}{2}\int (1+\cos 2\vartheta )d\vartheta ,x=\sin \vartheta$$$$=\frac{1}{2}({\sin }^{-1}\left(x\right)+x\sqrt{1-x^2})+C$$$$\Rightarrow \int x{\sin }^{-1}\left(x\right)dx=\frac{1}{2}{x}^2{\sin }^{-1}\left(x\right)-\frac{1}{4}x\sqrt{1-x^2}+\frac{1}{4}{\sin }^{-1}\left(x\right)+C$$

Commented by zakirullah last updated on 12/Jan/22

Terms of Service

Privacy Policy

Contact: info@tinkutara.com