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

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

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

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

\frac{3}{4} ln^{4} (x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int u^{3} d u

= 3 \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= 3 \cdot \frac{u ^{4}}{4}

Substitute back

u = ln (x)

= 3 \cdot \frac{ln ^{4} ( x )}{4}

Simplify

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

= \frac{3}{4} ln^{4} (x)

Add a constant to the solution

= \frac{3}{4} ln^{4} (x) + C

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