Question

$\int e^x\frac{(2-x^2)}{(1-x)\sqrt{1-x^2}}dx$.

Answer 1

$I=\int e^x\frac{2-x^2}{(1-x)\sqrt{1-x^2}}dx$

$=\int e^x\left(\frac{1+(1-x^2)}{(1-x)\sqrt{1-x^2}}\right)dx$

$=\int e^x[\frac{1-x^2}{(1-x)\sqrt{1-x^2}}+\frac{1}{(1-x)\sqrt{1-x^2}}]dx$

$=\int e^x\left[\right(\frac{(1+x)(1-x)}{(1-x)\sqrt{1+x}\sqrt{1-x}})+(\frac{1}{(1-x)\sqrt{1-x^2}}\left)\right]dx$

$=\int e^x(\sqrt{\frac{1+x}{1-x}}+\frac{1}{(1-x)\sqrt{1-x^2}})dx$

$I=\int e^x\sqrt{\frac{1+x}{1-x}}+c$ $(\int e^x(f\left(x\right)+f\left(x\right))=e^{2}f(x)+c)$