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

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

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integral of xln^2(x)

\int x 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

\int \frac{x}{( x ^{2} + 1 ) ^{\frac{23}{2}}} d x

x \to 10 - lim (ln (100 - x^{2}))

\frac{\partial}{\partial y} (2 y^{2})

x \to \infty + lim (i x)

\int_{0}^{\frac{π}{2}} 1

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