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

The integral from 0 to 3 of x^3e^{2x} is (69e^6+3)/8

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

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

\frac{69 e ^{6} + 3}{8}

Decimal

3479.94834 \dots

Solution steps

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

Apply Integration By Parts

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

\int \frac{3}{2} e^{2 x} x^{2} d x = \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))

= [\frac{1}{2} e^{2 x} x^{3} - \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))]_{0}^{3}

Compute the boundaries:

\frac{69 e ^{6}}{8} + \frac{3}{8}

= \frac{69 e ^{6}}{8} + \frac{3}{8}

Simplify

= \frac{69 e ^{6} + 3}{8}

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What is the integral from 0 to 3 of x^3e^{2x} ?
The integral from 0 to 3 of x^3e^{2x} is (69e^6+3)/8