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

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

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

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

x ln (ln (x)) + C

Solution steps

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

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

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

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

Simplify

= x ln (ln (x))

Add a constant to the solution

= x ln (ln (x)) + C

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