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

The integral of (ln(xe^x))/x is x+1/2 ln^2(x)+ln(x)-ln|x|-1+C

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integral of (ln(xe^x))/x

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

x + \frac{1}{2} ln^{2} (x) + ln (x) - ln ∣ x ∣ - 1 + C

Solution steps

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

Apply Integration By Parts

: \frac{1}{2} (x + 2 ln (x) - 2) - \int - \frac{x + 2 ln ( x ) - 2}{2 x} d x

= \frac{1}{2} (x + 2 ln (x) - 2) - \int - \frac{x + 2 ln ( x ) - 2}{2 x} d x

\int - \frac{x + 2 ln ( x ) - 2}{2 x} d x = - \frac{1}{2} x - \frac{1}{2} ln^{2} (x) + ln ∣ x ∣

= \frac{1}{2} (x + 2 ln (x) - 2) - (- \frac{1}{2} x - \frac{1}{2} ln^{2} (x) + ln ∣ x ∣)

Simplify

\frac{1}{2} (x + 2 ln (x) - 2) - (- \frac{1}{2} x - \frac{1}{2} ln^{2} (x) + ln ∣ x ∣) : x + \frac{1}{2} ln^{2} (x) + ln (x) - ln ∣ x ∣ - 1

= x + \frac{1}{2} ln^{2} (x) + ln (x) - ln ∣ x ∣ - 1

Add a constant to the solution

= x + \frac{1}{2} ln^{2} (x) + ln (x) - ln ∣ x ∣ - 1 + C

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What is the integral of (ln(xe^x))/x ?
The integral of (ln(xe^x))/x is x+1/2 ln^2(x)+ln(x)-ln|x|-1+C