What is the integral of 3z^2ln(z) ?

The integral of 3z^2ln(z) is z^3ln(z)-(z^3)/3+C

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integral of 3z^2ln(z)

\int 3 z^{2} ln (z) d z

z^{3} ln (z) - \frac{z ^{3}}{3} + C

Solution steps

\int 3 z^{2} ln (z) d z

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int z^{2} ln (z) d z

Apply Integration By Parts

: \frac{1}{3} z^{3} ln (z) - \int \frac{z ^{2}}{3} d z

= 3 (\frac{1}{3} z^{3} ln (z) - \int \frac{z ^{2}}{3} d z)

\int \frac{z ^{2}}{3} d z = \frac{z ^{3}}{9}

= 3 (\frac{1}{3} z^{3} ln (z) - \frac{z ^{3}}{9})

Simplify

3 (\frac{1}{3} z^{3} ln (z) - \frac{z ^{3}}{9}) : z^{3} ln (z) - \frac{z ^{3}}{3}

= z^{3} ln (z) - \frac{z ^{3}}{3}

Add a constant to the solution

= z^{3} ln (z) - \frac{z ^{3}}{3} + C

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What is the integral of 3z^2ln(z) ?
The integral of 3z^2ln(z) is z^3ln(z)-(z^3)/3+C