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

The integral of ((ln(3x))^2)/x is (ln^3(3x))/3+C

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integral of ((ln(3x))^2)/x

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

\frac{ln ^{3} ( 3 x )}{3} + C

Solution steps

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

Simplify

= \int \frac{ln ^{2} ( 3 x )}{x} d x

Apply u-substitution

: \int \frac{ln ^{2} ( u )}{u} d u

= \int \frac{ln ^{2} ( u )}{u} d u

Apply u-substitution

: \int v^{2} d v

= \int v^{2} d v

Apply the Power Rule

: \frac{v ^{3}}{3}

= \frac{v ^{3}}{3}

Substitute back

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

Add a constant to the solution

= \frac{ln ^{3} ( 3 x )}{3} + C

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What is the integral of ((ln(3x))^2)/x ?
The integral of ((ln(3x))^2)/x is (ln^3(3x))/3+C