What is the integral from 0 to 2 of x^2ln(3x) ?

The integral from 0 to 2 of x^2ln(3x) is 8/3 ln(6)-8/9

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integral from 0 to 2 of x^2ln(3x)

\int_{0}^{2} x^{2} ln (3 x) d x

\frac{8}{3} ln (6) - \frac{8}{9}

Decimal Notation

3.88913 \dots

Solution steps

\int_{0}^{2} x^{2} ln (3 x) d x

Apply Integration By Parts

: [\frac{1}{3} x^{3} ln (3 x) - \int \frac{x ^{2}}{3} d x]_{0}^{2}

= [\frac{1}{3} x^{3} ln (3 x) - \int \frac{x ^{2}}{3} d x]_{0}^{2}

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

= [\frac{1}{3} x^{3} ln (3 x) - \frac{x ^{3}}{9}]_{0}^{2}

Compute the boundaries:

\frac{8}{3} ln (6) - \frac{8}{9}

= \frac{8}{3} ln (6) - \frac{8}{9}

\int_{2}^{3} x^{2} e^{- x} d x

\int_{0}^{5} \frac{1}{5} d x

\int_{0}^{\infty} (\frac{1}{2})^{x} d x

\int_{- \infty}^{1} \frac{1}{x} d x

\int_{0}^{2} 1 - ∣ 1 - x ∣ d x

What is the integral from 0 to 2 of x^2ln(3x) ?
The integral from 0 to 2 of x^2ln(3x) is 8/3 ln(6)-8/9