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

The integral of ((ln(x))^3)/(2x) is 1/8 ln^4(x)+C

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

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

\frac{1}{8} ln^{4} (x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= \frac{1}{2} \cdot \int u^{3} d u

Apply the Power Rule

= \frac{1}{2} \cdot \frac{u ^{4}}{4}

Substitute back

u = ln (x)

= \frac{1}{2} \cdot \frac{ln ^{4} ( x )}{4}

Simplify

\frac{1}{2} \cdot \frac{ln ^{4} ( x )}{4} : \frac{1}{8} ln^{4} (x)

= \frac{1}{8} ln^{4} (x)

Add a constant to the solution

= \frac{1}{8} ln^{4} (x) + C

What is the integral of ((ln(x))^3)/(2x) ?
The integral of ((ln(x))^3)/(2x) is 1/8 ln^4(x)+C