Evaluate the integral: int frac{(x+1) dx}{x(1 | vedclass

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(A) Let $I = \int \frac{(x+1) dx}{x(1+xe^x)}$.Multiply the numerator and denominator by $e^x$:$I = \int \frac{(x+1)e^x dx}{xe^x(1+xe^x)}$.Let $u = xe^

Vedclass · Accepted Answer

$\log \left| \frac{xe^x}{1+xe^x} \right| + c$

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(A) Let $I = \int \frac{(x+1) dx}{x(1+xe^x)}$.Multiply the numerator and denominator by $e^x$:$I = \int \frac{(x+1)e^x dx}{xe^x(1+xe^x)}$.Let $u = xe^x$. Then $du = (e^x + xe^x) dx = e^x(1+x) dx$.Substituting these into the integral:$I = \int \frac{du}{u(1+u)}$.Using partial fractions: $\frac{1}{u(1+u)} = \frac{1}{u} - \frac{1}{1+u}$.$I = \int \left( \frac{1}{u} - \frac{1}{1+u} \right) du = \log|u| - \log|1+u| + c$.$I = \log \left| \frac{u}{1+u} \right| + c$.Substituting $u = xe^x$ back:$I = \log \left| \frac{xe^x}{1+xe^x} \right| + c$.

Evaluate the integral: $\int \frac{(x+1) dx}{x(1+xe^x)}$

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