Question

If $\int \frac{x^{e-1}+e^{x-1}}{x^e+e^x}dx=k\ln |x^e+e^x|+C,$ then the value of $4\ln k+7$ is equal to
(where $C$ is constant of integration)

Answer 1

$I=\int \frac{x^{e-1}+e^{x-1}}{x^e+e^x}dx=\frac{1}{e}\int \frac{e{x}^{e-1}+e^x}{x^e+e^x}dx$
putting $x^e+e^x=t$
$\Rightarrow (e{x}^{e-1}+e^x)dx=dt\therefore I=\frac{1}{e}\int \frac{1}{t}dt=\frac{1}{e}\ln |t|+C=\frac{1}{e}\ln |x^e+e^x|+C$
$\Rightarrow k=\frac{1}{e}\Rightarrow 4\ln \left(\frac{1}{e}\right)+7=-4+7=3$