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

The integral of (f(ln(x)))/x is f/2 ln^2(x)+C

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integral of (f(ln(x)))/x

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int u d u

= f \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= f \frac{u ^{2}}{2}

Substitute back

u = ln (x)

= f \frac{ln ^{2} ( x )}{2}

Simplify

f \frac{ln ^{2} ( x )}{2} : \frac{f}{2} ln^{2} (x)

= \frac{f}{2} ln^{2} (x)

Add a constant to the solution

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

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