# What is the integral of sqrt(ln(x))

1. Apr 13, 2015

### Emmanuel_Euler

what is the integral of sqrt(ln(x))
NOTE:(I do not need the final answer,I know it. I want to Know how to solve it.)

2. Apr 13, 2015

### robphy

We usually discourage such terse questions... especially without the poster any showing to attempts at the solution (as you've done in several recent posts).
Can you tell us your answer and how are approaching arriving at it?

3. Apr 13, 2015

### Delta²

Do the substitution y=lnx, and then the substitution y=z^2. You ll end up with an integral of the form $\int z^2e^{z^2}dz$ which if you look up at wolfram i believe you ll understand how it is calculated (at first glance the integral would be equal to $ze^{z^2}/2-f(z)$ where f(z) should be such that $f'(z)=e^{z^2}/2$). Such a function f(z) is the erfi function.

4. Apr 14, 2015

### Emmanuel_Euler

Look i substituted (x=e^(-t^2))
dx=-2t*e^(-t^2)dt

∫√(ln(e^(-t^2))*-2t*e^(-t^2)dt
-2∫√(-t^2)*t*e^(-t^2)dt

-2∫it^2*e^(-t^2)dt
When i arrived here i stopped.
but i used integral calculator to find the final answer.

½(√π*i* erf(i√(ln(x)))+2x√(ln(x)))+C

5. Apr 14, 2015

### HallsofIvy

Staff Emeritus
Do you understand what that means? Do you understand that almost all such functions have NO integral in terms of elementary functions?

6. Apr 14, 2015

### Emmanuel_Euler

yes i understand. thank you.