What is the integral from 0 to 1 of x8^x ?

The integral from 0 to 1 of x8^x is (24ln(2)-7)/(9ln^2(2))

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integral from 0 to 1 of x8^x

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

\frac{24 ln ( 2 ) - 7}{9 ln ^{2} ( 2 )}

Decimal

2.22834 \dots

Solution steps

\int_{0}^{1} x \cdot 8^{x} d x

Apply Integration By Parts

: [\frac{8 ^{x} x}{3 ln ( 2 )} - \int \frac{8 ^{x}}{3 ln ( 2 )} d x]_{0}^{1}

= [\frac{8 ^{x} x}{3 ln ( 2 )} - \int \frac{8 ^{x}}{3 ln ( 2 )} d x]_{0}^{1}

\int \frac{8 ^{x}}{3 ln ( 2 )} d x = \frac{8 ^{x}}{9 ln ^{2} ( 2 )}

= [\frac{8 ^{x} x}{3 ln ( 2 )} - \frac{8 ^{x}}{9 ln ^{2} ( 2 )}]_{0}^{1}

Compute the boundaries:

\frac{24 ln ( 2 ) - 7}{9 ln ^{2} ( 2 )}

= \frac{24 ln ( 2 ) - 7}{9 ln ^{2} ( 2 )}

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What is the integral from 0 to 1 of x8^x ?
The integral from 0 to 1 of x8^x is (24ln(2)-7)/(9ln^2(2))