What is the integral of (ln((4x)^3))/x ?

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

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integral of (ln((4x)^3))/x

\int \frac{ln ( ( 4 x ) ^{3} )}{x} d x

\frac{3}{2} (ln (x) + 2 ln (2))^{2} + C

Solution steps

\int \frac{ln ( ( 4 x ) ^{3} )}{x} d x

Simplify

\frac{ln ( ( 4 x ) ^{3} )}{x} : \frac{ln ( 64 x ^{3} )}{x}

= \int \frac{ln ( 64 x ^{3} )}{x} d x

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{ln ( u )}{u} d u

Apply u-substitution

: \int v d v

= \frac{1}{3} \cdot \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= \frac{1}{3} \cdot \frac{v ^{2}}{2}

Substitute back

= \frac{1}{3} \cdot \frac{ln ^{2} ( 64 x ^{3} )}{2}

Simplify

\frac{1}{3} \cdot \frac{ln ^{2} ( 64 x ^{3} )}{2} : \frac{3}{2} (ln (x) + 2 ln (2))^{2}

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

Add a constant to the solution

= \frac{3}{2} (ln (x) + 2 ln (2))^{2} + C

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