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

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

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

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

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

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

Apply u-substitution

: \int_{0}^{1} u^{2} d u

= \int_{0}^{1} u^{2} d u

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{0}^{1}

= [\frac{u ^{3}}{3}]_{0}^{1}

Compute the boundaries:

\frac{1}{3}

= \frac{1}{3}

x \to 0 lim (\frac{1}{x ( x + 1 )})

\int_{0}^{9} \frac{6}{( x - 1 ) ^{\frac{1}{3}}} d x

10 x

\int \frac{2}{3} d x

y = x^{2} + 7, (- 4, 23)

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