Question

integral x/((1-x^2)^(1/2)) dx

Answer 1

\[\int\limits \frac{x}{\sqrt{1-x^2}}dx ?\]

Answer 2

Make the substitution:
\[\zeta=1-x^2 \implies d \zeta=-2x dx \implies \frac{-1}{2} d \zeta =x dx\]
Replacing this the integral becomes:
\[\frac{-1}{2}\int\limits \frac{d \zeta}{\sqrt{\zeta}}=\frac{-1}{2}\int\limits \zeta^{\frac{-1}{2}}d \zeta=-\zeta^{\frac{1}{2}}+C=-\sqrt{1-x^2}+C\]

Answer 3

Here is the answer, does it make sense?