Question

Integrate $\frac{x}{\sqrt{1-x^2}}$ w.r.t $x$

Answer 1

$\int \frac{x}{\sqrt{1-x^2}}dx$

Let $x=\cos t⟹dx=\sin tdt$

$⟹\int \frac{\cos t}{\sqrt{1-{\cos }^{2}t}}\sin tdt$

$⟹\int \cos tdt=-\sin t$

$=-\sin {\cos }^{-1}x$

$=-\sin {\sin }^{-1}\sqrt{1-x^2}$

$=-\sqrt{1-x^2}$