What is the integral from 0 to 2 of x^2e^{3x} ?

The integral from 0 to 2 of x^2e^{3x} is (26e^6-2)/(27)

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integral from 0 to 2 of x^2e^{3x}

\int_{0}^{2} x^{2} e^{3 x} d x

\frac{26 e ^{6} - 2}{27}

Decimal Notation

388.41291 \dots

Solution steps

\int_{0}^{2} x^{2} e^{3 x} d x

Apply Integration By Parts

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

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

\int \frac{2}{3} e^{3 x} x d x = \frac{2}{27} (3 e^{3 x} x - e^{3 x})

= [\frac{1}{3} e^{3 x} x^{2} - \frac{2}{27} (3 e^{3 x} x - e^{3 x})]_{0}^{2}

Compute the boundaries:

\frac{26 e ^{6}}{27} - \frac{2}{27}

= \frac{26 e ^{6}}{27} - \frac{2}{27}

Simplify

= \frac{26 e ^{6} - 2}{27}

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What is the integral from 0 to 2 of x^2e^{3x} ?
The integral from 0 to 2 of x^2e^{3x} is (26e^6-2)/(27)