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Evaluate the integral \[\int \frac{e^x\left(1+x^{n-1}-x^n\right)}{(1-x^n)\sqrt{1-x^{2n}}}\,dx\].

1. \(\dfrac{e^x\sqrt{1-x^n}}{1-x^n}+C\)
2. \(\dfrac{e^x\sqrt{1+x^{2n}}}{1-x^n}+C\)
3. \(\dfrac{e^x\sqrt{1-x^{2n}}}{1-x^{2n}}+C\)
4. \(\dfrac{e^x\sqrt{1-x^{2n}}}{1-x^n}+C\)

Correct answer: \(\dfrac{e^x\sqrt{1+x^{2n}}}{1-x^n}+C\)

Solution

The correct option is derived from simplifying the integral using substitution and recognizing the relationship between the terms in the integrand. The presence of the square root and the exponential function suggests that the integral can be expressed in a form that combines these elements, leading to the correct expression.

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