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  4. The value of ∫ ex(1+nxn-1-x2n/(1-xn)√1-x2n)dx is

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

Integrals

Solution:

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