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

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

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

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

\frac{1}{6}

Decimal

0.16666 \dots

Solution steps

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

Apply u-substitution

: \int_{0}^{1} u^{5} d u

= \int_{0}^{1} u^{5} d u

Apply the Power Rule

: [\frac{u ^{6}}{6}]_{0}^{1}

= [\frac{u ^{6}}{6}]_{0}^{1}

Compute the boundaries:

\frac{1}{6}

= \frac{1}{6}

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