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

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

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

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

\frac{1}{8} ln^{2} (x) + C

Solution steps

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

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 ( x )}{x} d x

Apply u-substitution

= \frac{1}{4} \cdot \int u d u

Apply the Power Rule

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

Substitute back

u = ln (x)

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

Simplify

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

= \frac{1}{8} ln^{2} (x)

Add a constant to the solution

= \frac{1}{8} ln^{2} (x) + C

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