- #1

0kelvin

- 50

- 5

- Homework Statement
- Integral of ln(x) from 1 to infinite diverges. But how do I know if the 1/ln(x) will diverge too?

- Relevant Equations
- 1/ln(x)

Some functions have straight foward integrals, but they get complicated if you take the inverse of it. 1/f(x) for instance.

The primitive of 1/x is ln(x). In this case it's easy to check that the integral of 1/x or ln(x) from 1 to infinite diverges.

##\int_1^\infty (\ln(x))^n dx##

If n = 0, I have f(x) = 1. This cannot converge.

If n = 1, I have that the integral diverges.

If n < 0, then I have no idea except to let wolfram tell me.

If 1 < n < 0, the integral of ln(x) already diverges, taking the root of it just slows down a bit but still diverges.

The primitive of 1/x is ln(x). In this case it's easy to check that the integral of 1/x or ln(x) from 1 to infinite diverges.

##\int_1^\infty (\ln(x))^n dx##

If n = 0, I have f(x) = 1. This cannot converge.

If n = 1, I have that the integral diverges.

If n < 0, then I have no idea except to let wolfram tell me.

If 1 < n < 0, the integral of ln(x) already diverges, taking the root of it just slows down a bit but still diverges.