What is the integral of [ln(ln(x))+1/(ln(x))] ?

The integral of [ln(ln(x))+1/(ln(x))] is xln(ln(x))+C

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integral of [ln(ln(x))+1/(ln(x))]

\int [ln (ln (x)) + \frac{1}{ln ( x )}] d x

x ln (ln (x)) + C

Solution steps

\int ln (ln (x)) + \frac{1}{ln ( x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int ln (ln (x)) d x + \int \frac{1}{ln ( x )} d x

\int ln (ln (x)) d x = x ln (ln (x)) - Ei (ln (x))

\int \frac{1}{ln ( x )} d x = Ei (ln (x))

= x ln (ln (x)) - Ei (ln (x)) + Ei (ln (x))

Simplify

= x ln (ln (x))

Add a constant to the solution

= x ln (ln (x)) + C

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What is the integral of [ln(ln(x))+1/(ln(x))] ?
The integral of [ln(ln(x))+1/(ln(x))] is xln(ln(x))+C