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Integration by parts

  1. Sep 18, 2010 #1
    1. The problem statement, all variables and given/known data
    [tex] \int arctan(4t) [/tex]

    2. Relevant equations
    I know what the answer is to the problem but when i look at the solution i have no idea how they get from one step to the next.

    3. The attempt at a solution

    once we integrate by parts we get
    [tex] 1/4 U arctan(U) - 1/4 \int U/1+U^2 [/tex] where U= 4t

    from this they go to..

    [tex] 1/4 U arctan(U) - 1/8 \int 2U/1+U^2 [/tex]

    clearly a 2 was multiplied inside and a 1/2 on the outside of the integral, but why? and how? because the next step shown is..

    [tex] 1/4 (4t arctan(4t) - 1/8 ln 16t^2 + c [/tex]

    which is the answer..i get that 4t was substituted back in for U, but i dont understand how and why the 2 was put in the integral and the 1/2 outside of it, and how in the last step the 2U somehow is gone?
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 18, 2010 #2


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    Gold Member

    You should use parantheses to make your work clearer. Anyway, note that (1+u2)' = 2u. The two is multiplied (and divided) so that you can apply the substitution rule for integration.
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