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

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

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integral of (ln(y))^2y

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

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

Solution steps

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

Apply Integration By Parts

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

= \frac{1}{2} y^{2} ln^{2} (y) - \int y ln (y) d y

\int y ln (y) d y = \frac{1}{2} y^{2} ln (y) - \frac{y ^{2}}{4}

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

Simplify

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

Add a constant to the solution

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

\frac{x ^{2} + 4}{x ^{2} - 4}

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

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