What is the integral from 0 to 1 of x^2e^{6x} ?

The integral from 0 to 1 of x^2e^{6x} is (13e^6-1)/(108)

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integral from 0 to 1 of x^2e^{6x}

\int_{0}^{1} x^{2} e^{6 x} d x

\frac{13 e ^{6} - 1}{108}

Decimal Notation

48.55161 \dots

Solution steps

\int_{0}^{1} x^{2} e^{6 x} d x

Apply Integration By Parts

: [\frac{1}{6} e^{6 x} x^{2} - \int \frac{1}{3} e^{6 x} x d x]_{0}^{1}

= [\frac{1}{6} e^{6 x} x^{2} - \int \frac{1}{3} e^{6 x} x d x]_{0}^{1}

\int \frac{1}{3} e^{6 x} x d x = \frac{1}{108} (6 e^{6 x} x - e^{6 x})

= [\frac{1}{6} e^{6 x} x^{2} - \frac{1}{108} (6 e^{6 x} x - e^{6 x})]_{0}^{1}

Compute the boundaries:

\frac{13 e ^{6}}{108} - \frac{1}{108}

= \frac{13 e ^{6}}{108} - \frac{1}{108}

Simplify

= \frac{13 e ^{6} - 1}{108}

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What is the integral from 0 to 1 of x^2e^{6x} ?
The integral from 0 to 1 of x^2e^{6x} is (13e^6-1)/(108)