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

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

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integral of (ln(x))^2x

\int (ln (x))^{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 (ln (x))^{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

y^{^{''}} + 4 y = 3 csc^{2} (2 t)

\int \frac{x ^{2} + x - 2}{( x + 2 ) ( x - 5 ) ( x + 1 )} d x

\frac{\partial}{\partial x} (2 sin (x y))

f (x) = - x^{3} + 15 x^{2} - 48 x + 450

x \to 1 lim (- \frac{2}{x ^{2} - 1})

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