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

The integral of 1/(sqrt(ln(x))) is sqrt(pi)\erfi(sqrt(ln(x)))+C

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

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

π erfi (ln (x)) + C

Solution steps

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

Apply u-substitution

= \int 2 e^{u^{2}} d u

Take the constant out:

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

= 2 \cdot \int e^{u^{2}} d u

Use the nonelementary integral:

\int e^{u^{2}} d u = \frac{π}{2} erfi (u)

= 2 \cdot \frac{π}{2} erfi (u)

Substitute back

u = ln (x)

= 2 \cdot \frac{π}{2} erfi (ln (x))

Simplify

2 \cdot \frac{π}{2} erfi (ln (x)) : π erfi (ln (x))

= π erfi (ln (x))

Add a constant to the solution

= π erfi (ln (x)) + C

What is the integral of 1/(sqrt(ln(x))) ?
The integral of 1/(sqrt(ln(x))) is sqrt(pi)\erfi(sqrt(ln(x)))+C