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

The integral of ((ln(2x))^3)/x is (ln^4(2x))/4+C

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

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

\frac{ln ^{4} ( 2 x )}{4} + C

Solution steps

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

Simplify

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

Apply u-substitution

: \int \frac{ln ^{3} ( u )}{u} d u

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

Apply u-substitution

: \int v^{3} d v

= \int v^{3} d v

Apply the Power Rule

: \frac{v ^{4}}{4}

= \frac{v ^{4}}{4}

Substitute back

= \frac{ln ^{4} ( 2 x )}{4}

Add a constant to the solution

= \frac{ln ^{4} ( 2 x )}{4} + C

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What is the integral of ((ln(2x))^3)/x ?
The integral of ((ln(2x))^3)/x is (ln^4(2x))/4+C