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

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

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

\int_{1}^{3} \frac{2 ( ln ( x ) ) ^{2}}{x} d x

\frac{2 ln ^{3} ( 3 )}{3}

Decimal

0.88397 \dots

Solution steps

\int_{1}^{3} \frac{2 ( ln ( x ) ) ^{2}}{x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{1}^{3} \frac{( ln ( x ) ) ^{2}}{x} d x

Apply u-substitution

: \int_{0}^{l n (3)} u^{2} d u

= 2 \cdot \int_{0}^{l n (3)} u^{2} d u

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{0}^{l n (3)}

= 2 [\frac{u ^{3}}{3}]_{0}^{l n (3)}

Compute the boundaries:

\frac{ln ^{3} ( 3 )}{3}

= 2 \cdot \frac{ln ^{3} ( 3 )}{3}

Simplify

= \frac{2 ln ^{3} ( 3 )}{3}

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