Question

How do you find the integral `int_e^(e^4)dx/(x*sqrt(ln(x)))dx` ?

Answer 1

Since dx has been placed twice, I'm going to assume that the equation should read `int_e^(e^4)dx/(x*sqrt(ln(x)))`. In this case, a good way to find the integral is by substitution, letting `u = ln(x)`.

To integrate something by substitution (also known as the change-of-variable rule), we need to select a function `u` so that its derivative also forms part of the original equation. (For example, when we try to antidifferentiate `tan(x)` we can say that `tan(x) = sin(x)/cos(x)` and select `u = cos(x)`. The derivative of `u` is also within the original equation.)

To find the integral of your function, we do the following:

1. Let `u = ln(x)`, then `(du)/dx = 1/x` - this is a standard derivative.
2. Substitute these two new functions into the equation:
   `int_e^(e^4)dx/(x*sqrt(ln(x))) = int_e^(e^4)1/(x*u)dx = int_e^(e^4)1/u*(du)/dxdx`
3. Find new terminals - this is a crucial step, because we're changing the variable!
   `"When "x = e, u = ln(e) = 1 " and when "x=e^4, u = ln(e^4)=4`
4. Substitute these new terminals in, and "cancel out" the two `dx` terms:
   `int_e^(e^4)1/u*(du)/dxdx = int_1^4(du)/u = int_1^4u^(-1)du`
5. Integrate normally. The answer you get for this "new" integral will be exactly the same answer as the original integral.