What is the integral from 1 to e of x^9ln(x) ?

The integral from 1 to e of x^9ln(x) is (9e^{10}+1)/(100)

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integral from 1 to e of x^9ln(x)

\int_{1}^{e} x^{9} ln (x) d x

\frac{9 e ^{10} + 1}{100}

Decimal

1982.39192 \dots

Solution steps

\int_{1}^{e} x^{9} ln (x) d x

Apply Integration By Parts

: [\frac{1}{10} (x^{10} ln (x) - 10 \cdot \int \frac{x ^{9}}{10} d x)]_{1}^{e}

= [\frac{1}{10} (x^{10} ln (x) - 10 \cdot \int \frac{x ^{9}}{10} d x)]_{1}^{e}

\int \frac{x ^{9}}{10} d x = \frac{x ^{10}}{100}

= [\frac{1}{10} (x^{10} ln (x) - 10 \cdot \frac{x ^{10}}{100})]_{1}^{e}

10 \cdot \frac{x ^{10}}{100} = \frac{x ^{10}}{10}

= [\frac{1}{10} (x^{10} ln (x) - \frac{x ^{10}}{10})]_{1}^{e}

Compute the boundaries:

\frac{9 e ^{10}}{100} + \frac{1}{100}

= \frac{9 e ^{10}}{100} + \frac{1}{100}

Simplify

= \frac{9 e ^{10} + 1}{100}

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What is the integral from 1 to e of x^9ln(x) ?
The integral from 1 to e of x^9ln(x) is (9e^{10}+1)/(100)