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Question Number 99114 by  M±th+et+s last updated on 18/Jun/20

$$calculate:$$$$\int \sqrt{x}{sinh}^{-1}\left(x\right)dx$$$$where{sinh}^{-1}\left(x\right)istheinversehyperbolic$$$$sinefunction$$$$$$$$$$

Answered by Rio Michael last updated on 19/Jun/20

$$\mathrm{By}\mathrm{parts}$$$$\mathrm{let}u={\sinh }^{-1}x\mathrm{and}{v}^{\prime }=\sqrt{x}$$$$\Rightarrow u^{\prime }=\frac{1}{\sqrt{1+x^2}}\mathrm{and}v=\frac{2x^{\frac{3}{2}}}{3}$$$$\Rightarrow \int \sqrt{x}\sinh {}^{-1}xdx=\left[\frac{2x^{3/2}}{3}{\sinh }^{-1}x\right]-\frac{2}{3}\int \frac{x^{3/2}}{\sqrt{x^2+1}}dx$$$$\mathrm{to}\mathrm{be}\mathrm{continued}.....$$

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