What is the integral of ln(xe^x) ?

The integral of ln(xe^x) is xln(x)+x^2-x-(x^2)/2+C

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integral of ln(xe^x)

\int ln (x e^{x}) d x

x ln (x) + x^{2} - x - \frac{x ^{2}}{2} + C

Solution steps

\int ln (x e^{x}) d x

Apply Integration By Parts

: x (ln (x) + x) - \int 1 + x d x

= x (ln (x) + x) - \int 1 + x d x

\int 1 + x d x = x + \frac{x ^{2}}{2}

= x (ln (x) + x) - (x + \frac{x ^{2}}{2})

Simplify

x (ln (x) + x) - (x + \frac{x ^{2}}{2}) : x ln (x) + x^{2} - x - \frac{x ^{2}}{2}

= x ln (x) + x^{2} - x - \frac{x ^{2}}{2}

Add a constant to the solution

= x ln (x) + x^{2} - x - \frac{x ^{2}}{2} + C

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What is the integral of ln(xe^x) ?
The integral of ln(xe^x) is xln(x)+x^2-x-(x^2)/2+C