Question

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

Answer 1

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

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

Let $(e^x+x^e)=z$

So, $(e^x+{ex}^{e-1})dx=dz$

So, $I=\frac{1}{e}\int \frac{1}{z}dz=\frac{1}{e}log\left|z\right|+C$

$I=\frac{1}{e}\log |e^x+x^e|+C$