What is the integral from 1 to e of x^8ln(x) ?

The integral from 1 to e of x^8ln(x) is (8e^9+1)/(81)

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integral from 1 to e of x^8ln(x)

\int_{1}^{e} x^{8} ln (x) d x

\frac{8 e ^{9} + 1}{81}

Decimal

800.31693 \dots

Solution steps

\int_{1}^{e} x^{8} ln (x) d x

Apply Integration By Parts

= [\frac{1}{9} (x^{9} ln (x) - 9 \cdot \int \frac{x ^{8}}{9} d x)]_{1}^{e}

\int \frac{x ^{8}}{9} d x = \frac{x ^{9}}{81}

= [\frac{1}{9} (x^{9} ln (x) - 9 \cdot \frac{x ^{9}}{81})]_{1}^{e}

9 \cdot \frac{x ^{9}}{81} = \frac{x ^{9}}{9}

= [\frac{1}{9} (x^{9} ln (x) - \frac{x ^{9}}{9})]_{1}^{e}

Compute the boundaries:

\frac{8 e ^{9}}{81} + \frac{1}{81}

= \frac{8 e ^{9}}{81} + \frac{1}{81}

Simplify

= \frac{8 e ^{9} + 1}{81}

What is the integral from 1 to e of x^8ln(x) ?
The integral from 1 to e of x^8ln(x) is (8e^9+1)/(81)