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

The integral of (ln(ln(x)))/(2x) is 1/2 (ln(x)ln(ln(x))-ln(x))+C

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

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

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

Solution steps

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

Apply u-substitution

: \int ln (u) d u

= \frac{1}{2} \cdot \int ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= \frac{1}{2} (u ln (u) - \int 1 d u)

\int 1 d u = u

= \frac{1}{2} (u ln (u) - u)

Substitute back

u = ln (x)

= \frac{1}{2} (ln (x) ln (ln (x)) - ln (x))

Add a constant to the solution

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

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