How do we integrate 1/lnx.
Don't waste a lot of time trying to figure this one out.
The integral is a specially defined function, li x.
See
http://en.wikipedia.org/wiki/Logarithmic_integral_function
To integrate the function 1/ln(x), you can use a technique called substitution. Here's the step-by-step process to integrate this function:
1. Let's start by making a substitution. We'll substitute u = ln(x), which means du/dx = 1/x, or dx = du/x.
2. Now we need to rewrite the integral in terms of u. The function 1/ln(x) becomes 1/u, and dx becomes du/x.
3. Substitute these new expressions into the integral. The integral becomes:
∫ (1/u) * (du/x)
4. Simplify the expression:
∫ (1/u) * (du/x) = ∫ (1/u) * (1/x) * du
5. We can rearrange the terms and pull out the constant factors:
∫ (1/u) * (1/x) * du = (1/x) * ∫ (1/u) * du
6. Now we need to integrate the remaining function. The integral of 1/u with respect to u is ln|u|. So, the integral becomes:
(1/x) * ln|u| + C
7. Substitute back the original expression for u, which is ln(x):
(1/x) * ln|u| + C = (1/x) * ln|ln(x)| + C
And there you have it. The integration of 1/ln(x) is (1/x) * ln|ln(x)| + C, where C is the constant of integration.