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Using u substitution, which of the following is equivalent to this integral?

  1. May 3, 2010 #1
    1. The problem statement, all variables and given/known data

    Using the u substutituion u = 2x + 1, ∫(2x + 1)1/2dx (when x goes from 0 to 2) is equivalent to?

    Answer: (1/2)*∫(u)1/2du (when x goes from 1 to 5)

    2. Relevant equations

    3. The attempt at a solution

    If u is 2x + 1, then du = 2dx. Thus, I get

    (1/2)*∫(u)1/2du (when x goes from 0 to 2)​

    The only problem is where did the x goes from 1 to 5 come from? Did I do something wrong, or is the AP packet wrong?
  2. jcsd
  3. May 3, 2010 #2
    For a question like this, when you use substitution to solve a definite integral and have it in a form with u in it, you need to have the two limits in terms of u as well, not x.

    So in this example, the limits are 0 to 2 and you know that u = 2x + 1. Just substitute in x = 0 and x = 2, and it gives you the new limits of 1 and 5.
    Last edited: May 3, 2010
  4. May 3, 2010 #3
    Oh, yes, you are right. I forgot that in order for my answer to be right, it had to be (1/2)*∫(2x + 1)1/2du (when x goes from 0 to 2).

  5. May 3, 2010 #4


    Staff: Mentor

    That should be (1/2)*∫(2x + 1)1/2dx.

    It's not necessary to change the limits of integration when you do a substitution. You can work with the indefinite integral, make the substitution, get your antiderivative, undo your substitution, and then evaluate your antiderivative at the original limits.
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