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

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

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integral of 3x^2ln(x)

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

Simplify

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

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

Add a constant to the solution

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

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