What is the integral from 0 to 1 of x^4e^{8x} ?

The integral from 0 to 1 of x^4e^{8x} is (331e^8-3)/(4096)

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integral from 0 to 1 of x^4e^{8x}

\int_{0}^{1} x^{4} e^{8 x} d x

\frac{331 e ^{8} - 3}{4096}

Decimal

240.89211 \dots

Solution steps

\int_{0}^{1} x^{4} e^{8 x} d x

Apply Integration By Parts

= [\frac{1}{8} e^{8 x} x^{4} - \int \frac{1}{2} e^{8 x} x^{3} d x]_{0}^{1}

\int \frac{1}{2} e^{8 x} x^{3} d x = \frac{1}{8192} (512 e^{8 x} x^{3} - 3 (64 e^{8 x} x^{2} - 2 (8 e^{8 x} x - e^{8 x})))

= [\frac{1}{8} e^{8 x} x^{4} - \frac{1}{8192} (512 e^{8 x} x^{3} - 3 (64 e^{8 x} x^{2} - 2 (8 e^{8 x} x - e^{8 x})))]_{0}^{1}

Simplify

[\frac{1}{8} e^{8 x} x^{4} - \frac{1}{8192} (512 e^{8 x} x^{3} - 3 (64 e^{8 x} x^{2} - 2 (8 e^{8 x} x - e^{8 x})))]_{0}^{1} : [\frac{1}{4096} e^{8 x} (512 x^{4} - 256 x^{3} + 96 x^{2} - 24 x + 3)]_{0}^{1}

= [\frac{1}{4096} e^{8 x} (512 x^{4} - 256 x^{3} + 96 x^{2} - 24 x + 3)]_{0}^{1}

Compute the boundaries:

\frac{331 e ^{8}}{4096} - \frac{3}{4096}

= \frac{331 e ^{8}}{4096} - \frac{3}{4096}

Simplify

= \frac{331 e ^{8} - 3}{4096}

What is the integral from 0 to 1 of x^4e^{8x} ?
The integral from 0 to 1 of x^4e^{8x} is (331e^8-3)/(4096)