Evaluate $\int \frac{x+1}{x(1+xe^x)^2} dx$.

Question

Professor PC Thomas Classes · Answer

Multiply numerator and denominator by $e^x$: $\int \frac{e^x(x+1)}{xe^x(1+xe^x)^2} dx$. Let $t = xe^x$, then $dt = e^x(x+1)dx$. The integral reduces to $\int \frac{dt}{t(1+t)^2}$. Using partial fractions: $\frac{1}{t(1+t)^2} = \frac{1}{t} - \frac{1}{1+t} - \frac{1}{(1+t)^2}$. Integrating gives $\ln|t| - \ln|1+t| + \frac{1}{1+t} + C$. Substituting back $t = xe^x$, we get $\ln\left|\frac{xe^x}{1+xe^x}\right| + \frac{1}{1+xe^x} + C$.