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

1 Answer
Sep 6, 2014

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.