What is the integral of x(ln|x|)^2 ?

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

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integral of x(ln|x|)^2

\int x (ln ∣ x ∣)^{2} d x

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

Solution steps

\int x (ln ∣ x ∣)^{2} d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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