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

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

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integral of z(ln(z))^2

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial x} ((x^{2} y + 4 y) e^{2 x y})

\int \frac{( x ^{2} + 1 )}{5 cosh ^{2} ( x ^{3} + 3 x )} d x

\frac{\partial}{\partial y} (x^{2} + y^{2} - z)

\frac{\partial}{\partial y} (cos (x^{2} + y^{2}))

\int (3 x + 5) cos (\frac{x}{4}) d x

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