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

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

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

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

\frac{t ^{3}}{3}

Solution steps

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

Apply u-substitution

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

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

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{0}^{t}

= [\frac{u ^{3}}{3}]_{0}^{t}

Compute the boundaries:

\frac{t ^{3}}{3}

= \frac{t ^{3}}{3}

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