Question

Prove that $I=\int \frac{e^x(1+n{x}^{n-1}-x^{2n})}{(1-x^n\sqrt{1-x^{2n}})}dx=e^x\sqrt{\frac{1+x^n}{1-x^n}}$.

Answer 1

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

$f\left(x\right)=\frac{(1-x^n)(1+x^n)}{(1-x^n)\sqrt{(1-x^n)(1+x^n)}}$

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

$\Rightarrow I=e^x\sqrt{\frac{1+x^n}{1-x^n}}+C$