What is the integral from 1 to e of x^5ln(x) ?

The integral from 1 to e of x^5ln(x) is (5e^6+1)/(36)

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integral from 1 to e of x^5ln(x)

\int_{1}^{e} x^{5} ln (x) d x

\frac{5 e ^{6} + 1}{36}

Decimal Notation

56.05955 \dots

Solution steps

\int_{1}^{e} x^{5} ln (x) d x

Apply Integration By Parts

: [\frac{1}{6} (x^{6} ln (x) - 6 \cdot \int \frac{x ^{5}}{6} d x)]_{1}^{e}

= [\frac{1}{6} (x^{6} ln (x) - 6 \cdot \int \frac{x ^{5}}{6} d x)]_{1}^{e}

\int \frac{x ^{5}}{6} d x = \frac{x ^{6}}{36}

= [\frac{1}{6} (x^{6} ln (x) - 6 \cdot \frac{x ^{6}}{36})]_{1}^{e}

6 \cdot \frac{x ^{6}}{36} = \frac{x ^{6}}{6}

= [\frac{1}{6} (x^{6} ln (x) - \frac{x ^{6}}{6})]_{1}^{e}

Compute the boundaries:

\frac{5 e ^{6}}{36} + \frac{1}{36}

= \frac{5 e ^{6}}{36} + \frac{1}{36}

Simplify

= \frac{5 e ^{6} + 1}{36}

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What is the integral from 1 to e of x^5ln(x) ?
The integral from 1 to e of x^5ln(x) is (5e^6+1)/(36)