What is the integral from 1 to 9 of (3(ln(x))^2)/x ?

The integral from 1 to 9 of (3(ln(x))^2)/x is 8ln^3(3)

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integral from 1 to 9 of (3(ln(x))^2)/x

\int_{1}^{9} \frac{3 ( ln ( x ) ) ^{2}}{x} d x

8 ln^{3} (3)

Decimal

10.60775 \dots

Solution steps

\int_{1}^{9} \frac{3 ( ln ( x ) ) ^{2}}{x} d x

Take the constant out:

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

= 3 \cdot \int_{1}^{9} \frac{( ln ( x ) ) ^{2}}{x} d x

Apply u-substitution

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

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

Apply the Power Rule

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

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

Compute the boundaries:

\frac{8 ln ^{3} ( 3 )}{3}

= 3 \cdot \frac{8 ln ^{3} ( 3 )}{3}

Simplify

= 8 ln^{3} (3)

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What is the integral from 1 to 9 of (3(ln(x))^2)/x ?
The integral from 1 to 9 of (3(ln(x))^2)/x is 8ln^3(3)