What is the integral from 1 to e^2 of ((ln(x))^5)/x ?

The integral from 1 to e^2 of ((ln(x))^5)/x is 32/3

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integral from 1 to e^2 of ((ln(x))^5)/x

\int_{1}^{e^{2}} \frac{( ln ( x ) ) ^{5}}{x} d x

\frac{32}{3}

Decimal

10.66666 \dots

Solution steps

\int_{1}^{e^{2}} \frac{( ln ( x ) ) ^{5}}{x} d x

Apply u-substitution

: \int_{0}^{2} u^{5} d u

= \int_{0}^{2} u^{5} d u

Apply the Power Rule

: [\frac{u ^{6}}{6}]_{0}^{2}

= [\frac{u ^{6}}{6}]_{0}^{2}

Compute the boundaries:

\frac{32}{3}

= \frac{32}{3}

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What is the integral from 1 to e^2 of ((ln(x))^5)/x ?
The integral from 1 to e^2 of ((ln(x))^5)/x is 32/3