∫$$sin$$−$$1$$⁡$$x1$$−$$x23/2dx$$ is equal to

Question

∫$$sin$$−$$1$$⁡$$x1$$−$$x23/2dx$$ is equal to

Infinity Learn · Accepted Answer

Let $$I=$$∫$$sin$$−$$1$$⁡$$x1$$−$$x21$$−$$x2dxPut$$⁡$$sin$$−$$1$$⁡$$x=t$$⇒$$11$$−$$x2=dtdx$$⇒$$dx=1$$−$$x2dt$$∴$$I=$$∫$$t1$$−$$sin2$$⁡$$t1$$−$$x2$$⋅$$1$$−$$x2dt=$$∫$$t1$$−$$sin2$$⁡$$tdt=$$∫  $$tsec2$$⁡dt ∵$$1$$−$$sin2$$⁡$$t=cos2$$⁡$$t=t$$∫$$sec2$$⁡tdt−∫ddtt∫$$sec2$$⁡tdtdt                                                               [integration by parts] $$=ttan$$⁡t−∫$$tan$$⁡$$tdt=ttan$$⁡$$t+log$$⁡$$cos$$⁡$$t+C=$$$$sin$$−$$1$$⁡$$xx1$$−$$x2+log$$⁡$$1$$−$$x2+C$$ [fromfigure $$tan$$⁡$$t=x1$$−$$x2$$ and $$cos$$⁡$$t=1$$−$$x2$$ ]

$\int \frac{\sin^{- 1} x}{{(1 - x^{2})}^{3 / 2}} dx$ is equal to

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∫sin−1⁡x1−x23/2dx is equal to

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$\int \frac{\sin^{- 1} x}{{(1 - x^{2})}^{3 / 2}} dx$ is equal to