Question

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

• A. $e^x\sqrt{\frac{1+x}{1-x}}+c$
• B. $e^x\sqrt{\frac{x}{1-x}}+c$
• C. $e^x\sqrt{\frac{1-x}{1+x}}+c$
• D. none of these

Answer 1

The correct option is A $e^x\sqrt{\frac{1+x}{1-x}}+c$

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

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

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

$=e^x\sqrt{\frac{1+x}{1-x}}+c$