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

The integral of (ln^3(x^4))/x is 16ln^4(x)+C

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

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

16 ln^{4} (x) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{ln ^{3} ( u )}{u} d u

Apply u-substitution

: \int v^{3} d v

= \frac{1}{4} \cdot \int v^{3} d v

Apply the Power Rule

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

= \frac{1}{4} \cdot \frac{v ^{4}}{4}

Substitute back

= \frac{1}{4} \cdot \frac{ln ^{4} ( x ^{4} )}{4}

Simplify

\frac{1}{4} \cdot \frac{ln ^{4} ( x ^{4} )}{4} : 16 ln^{4} (x)

= 16 ln^{4} (x)

Add a constant to the solution

= 16 ln^{4} (x) + C

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What is the integral of (ln^3(x^4))/x ?
The integral of (ln^3(x^4))/x is 16ln^4(x)+C