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

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

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integral of ln^2(x)x

\int ln^{2} (x) x d x

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

Solution steps

\int ln^{2} (x) x d x

Apply Integration By Parts

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

Simplify

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

Add a constant to the solution

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

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