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

The integral of ((ln(x))^3)/(3x) is 1/12 ln^4(x)+C

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

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

\frac{1}{12} ln^{4} (x) + C

Solution steps

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

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 ( x ) ) ^{3}}{x} d x

Apply u-substitution

: \int u^{3} d u

= \frac{1}{3} \cdot \int u^{3} d u

Apply the Power Rule

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

= \frac{1}{3} \cdot \frac{u ^{4}}{4}

Substitute back

u = ln (x)

= \frac{1}{3} \cdot \frac{ln ^{4} ( x )}{4}

Simplify

\frac{1}{3} \cdot \frac{ln ^{4} ( x )}{4} : \frac{1}{12} ln^{4} (x)

= \frac{1}{12} ln^{4} (x)

Add a constant to the solution

= \frac{1}{12} ln^{4} (x) + C

n = 0 \sum \infty \frac{n x ^{n}}{n + 8}

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\int e x sin (x) d x

\frac{x}{2}

What is the integral of ((ln(x))^3)/(3x) ?
The integral of ((ln(x))^3)/(3x) is 1/12 ln^4(x)+C