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Evaluate the integral ∫ (e^x (1+xⁿ⁻¹-xⁿ))/((1-xⁿ)√(1-x²ⁿ)) dx.

1. (e^x√(1-xⁿ))/(1-xⁿ)+C
2. (e^x√(1+x²ⁿ))/(1-xⁿ)+C
3. (e^x√(1-x²ⁿ))/(1-x²ⁿ)+C
4. (e^x√(1-x²ⁿ))/(1-xⁿ)+C

Correct answer: (e^x√(1-x²ⁿ))/(1-xⁿ)+C

Solution

The integrand has the structure e^x[f(x)+f'(x)] with f(x)=sqrt(1-x^(2n))/(1-x^n). Therefore the integral is e^x*sqrt(1-x^(2n))/(1-x^n)+C.

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