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Question Number 91996 by student work last updated on 04/May/20

$$\int {\left(\mathrm{arcsinx}\right)}^2\mathrm{dx}=?$$

Commented by jagoll last updated on 04/May/20

$$\mathrm{let}{\sin }^{-1}\left(\mathrm{x}\right)=\mathrm{t}$$$$\mathrm{x}=\sin \mathrm{t}\Rightarrow \mathrm{dx}=\cos \mathrm{t}\mathrm{dt}$$$$\int {\mathrm{t}}^2\cos \mathrm{t}\mathrm{dt}=$$$${\mathrm{t}}^2\sin \mathrm{t}+2\mathrm{t}\cos \mathrm{t}-2\sin \mathrm{t}+\mathrm{c}$$$$=\mathrm{x}{\left({\sin }^{-1}\left(\mathrm{x}\right)\right)}^2+2\sqrt{1-{\mathrm{x}}^2}\left({\sin }^{-1}\left(\mathrm{x}\right)\right)-2\mathrm{x}+\mathrm{c}$$

Commented by jagoll last updated on 04/May/20

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