What is the integral from 1 to e of x^2ln(x^3) ?

The integral from 1 to e of x^2ln(x^3) is (2e^3+1)/3

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integral from 1 to e of x^2ln(x^3)

\int_{1}^{e} x^{2} ln (x^{3}) d x

\frac{2 e ^{3} + 1}{3}

Decimal

13.72369 \dots

Solution steps

\int_{1}^{e} x^{2} ln (x^{3}) d x

Apply Integration By Parts

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

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

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

= [\frac{1}{3} (x^{3} ln (x^{3}) - 3 \cdot \frac{x ^{3}}{3})]_{1}^{e}

Simplify

[\frac{1}{3} (x^{3} ln (x^{3}) - 3 \cdot \frac{x ^{3}}{3})]_{1}^{e} : [\frac{1}{3} (3 x^{3} ln (x) - x^{3})]_{1}^{e}

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

Compute the boundaries:

\frac{2 e ^{3}}{3} + \frac{1}{3}

= \frac{2 e ^{3}}{3} + \frac{1}{3}

Simplify

= \frac{2 e ^{3} + 1}{3}

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What is the integral from 1 to e of x^2ln(x^3) ?
The integral from 1 to e of x^2ln(x^3) is (2e^3+1)/3