What is the integral of xln|x| ?

The integral of xln|x| is 1/2 x^2ln|x|-(x^2)/4+C

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integral of xln|x|

\int x ln ∣ x ∣ d x

\frac{1}{2} x^{2} ln ∣ x ∣ - \frac{x ^{2}}{4} + C

Solution steps

\int x ln ∣ x ∣ d x

Apply Integration By Parts

: \frac{1}{2} x^{2} ln ∣ x ∣ - \int \frac{x}{2} d x

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= \frac{1}{2} x^{2} ln ∣ x ∣ - \frac{x ^{2}}{4}

Add a constant to the solution

= \frac{1}{2} x^{2} ln ∣ x ∣ - \frac{x ^{2}}{4} + C

x \to 8 lim (ln (9 - x))

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\int e^{\frac{1}{4} x} d x

x \to - \infty lim (e^{x} + 1)

\int (x^{2} - 12)^{5} (2 x) d x

What is the integral of xln|x| ?
The integral of xln|x| is 1/2 x^2ln|x|-(x^2)/4+C