How do you find the integral e4edxxln(x)dx ?

1 Answer
Sep 6, 2014

Since dx has been placed twice, I'm going to assume that the equation should read e4edxxln(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 dudx=1x - this is a standard derivative.
  2. Substitute these two new functions into the equation:
    e4edxxln(x)=e4e1xudx=e4e1ududxdx
  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=e4,u=ln(e4)=4
  4. Substitute these new terminals in, and "cancel out" the two dx terms:
    e4e1ududxdx=41duu=41u1du
  5. Integrate normally. The answer you get for this "new" integral will be exactly the same answer as the original integral.