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

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

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integral of x*ln^2(x)

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

Apply Integration By Parts

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

= \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

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

\int \frac{y}{3 - y} d y

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