Question

Evaluate $\int \frac{e^{x-1}+x^{e-1}}{e^x+x^e}dx$

Answer 1

$\int \frac{e^{x-1}+x^{e-1}}{e^x+x^e}dx=\int \frac{\frac{e^x}{e}+x^{e-1}}{e^x+x^e}dx=\int \frac{e^x+e{x}^{e-1}}{e^x+x^e}\frac{dx}{e}=\int \frac{d(e^x+x^e)}{e^x+x^e}\times \frac{1}{e}=\frac{1}{e}\ln (e^x+x^e)$