Question

Solve:

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

Answer 1

Given the integral,

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

Let us assume,

$u=x^e+e^x\Rightarrow \frac{du}{dx}=e^x+e{x}^{e-1}\Rightarrow du=(e^x+e{x}^{e-1})dx$

Substituting these values in the integral we get,

$\int \frac{e^{x-1}+x^{e-1}}{x^e+e^x}dx=\int \frac{e^{-1}}{u}du=e^{-1}\int \frac{1}{u}du=e^{-1}\ln \left(u\right)=e^{-1}\ln (x^e+e^x)\therefore \int \frac{e^{x-1}+x^{e-1}}{x^e+e^x}dx=e^{-1}\ln \left(|x^e+e^x|\right)+C.$