Question

The value of $\int e^x\frac{1+n{x}^{n-1}-x^{2n}}{\left(1-x^n\right)\sqrt{1-x^{2n}}}dx$ is

• A. $e^x\frac{\sqrt{1-x^n}}{1-x^n}+C$
• B. $e^x\frac{\sqrt{1+x^{2n}}}{1-x^{2n}}+C$
• C. $e^x\frac{\sqrt{1-x^{2n}}}{1-x^{2n}}+C$
• D. $e^x\frac{\sqrt{1-x^{2n}}}{1-x^n}+C$

Answer 1

Answer: (D)

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