<s>[INST] Evaluation of given indefinite integral Evaluate the given integral
$$\int e^x \bigg[\frac{2-x^2}{(1-x)\sqrt{1-x^2}} \bigg]dx$$
I was trying to convert it to $\int e^x (f(x)+f'(x))dx=e^x \cdot f(x)+C$ but did not succeed in algebraic manipulations. Could someone hint me to something so that I could proceed?
 [/INST] $$\begin{aligned} 
\frac{2-x^2}{(1-x)\sqrt{1-x^2}} 
&=\frac{1}{(1-x)\sqrt{1-x^2}}+\frac{1-x^2}{(1-x)\sqrt{1-x^2}} \\
&=\frac{1}{(1-x)\sqrt{1-x^2}}+\frac{\sqrt{1-x^2}}{(1-x)} \\
&=\frac{1}{(1-x)\sqrt{1-x^2}}+\frac{\sqrt{1+x}}{\sqrt{1-x}} \\
\end{aligned}$$
The only thing left is to notice that,I'll leave that to you.
$$\left(\frac{\sqrt{1+x}}{\sqrt{1-x}}\right)'=\frac{1}{(1-x)\sqrt{1-x^2}}$$
</s>