$\int 1 / (\ln x) dx$

By Andrew Hodges (P4403) on Tuesday, May 8, 2001 - 04:35 pm :

Does the integral of

1 / (\ln x)

have a closed form? If so what is it? Certain functions, such as

e^{- \frac{1}{2} x^{2}}

can not be integrated algebraically - only a Taylor series for the integrated function can be derived. What determines whether a function in general can be integrated to give a closed form answer?

By Dan Goodman (Dfmg2) on Tuesday, May 8, 2001 - 06:27 pm :

The integral of 1/log(x) (called Li(x)) doesn't have a closed form solution in terms of elementary functions. Interestingly, it's a very good estimate of the number of primes less than or equal to x, but that's another matter. In general, it's quite tricky to show what functions are or are not integrable in terms of elementary functions. The field of "differential Galois theory" actually answers your question, searching on Google brings up some good links.