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

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

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integral of x(ln(x))^2

integral

x (ln (x))^{2}

\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 (x))^{2} 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

e^{2} x \cdot (e^{e})^{x}

\frac{( 2 t ^{2} - 1 )}{t}

\frac{1 - e ^{- x}}{x}

\frac{x ^{3}}{( x + 1 ) ^{3}}

2^{x} \cdot e^{2} x

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