Question

$\int e^x\frac{(1+n.x^{n-1}-x^{2n})}{(1-x^n)\sqrt{1-x^{2n}}}$ dx is equal to

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

Answer 1

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

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 $T=\frac{1-x^{2n}}{(1-x^n)\sqrt{1-x^{2n}}}=\sqrt{\frac{1-x^n}{1-x^n}}$
$\frac{dT}{dx}=\frac{{nx}^{n-1}}{(1-x^n)\sqrt{1-x^{2n}}}$
$\therefore I=e^{x}T=e^x\sqrt{\frac{1+x^n}{1-x^n}}$