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Hard definite integration by parts question, need help

  1. Jun 24, 2012 #1
    1. The problem statement, all variables and given/known data

    Integral, from 0 to 1, y/(e^2y)

    2. Relevant equations
    Use integration by parts


    3. The attempt at a solution

    I need integration by parts. The answer is 1/4-3/5e^-2. But I got 1/2 and it all makes sense to me, so tell me what I got wrong.

    I put u=e^2y
    du=2e^2y

    dv= 1/e^2y dy (did u substitution on 2y here)
    v=1/2 * ln(e^2y) (here I canceled ln and e, so left with 2y)
    v=1/2 * 2y = y

    Plugged them into integration by parts formula
    and got y^2-(y^2)/2. Plugged in 1 and 0.
    And I got 1-1/2 = 1/2. How is this wrong?? I don't understand.
     
    Last edited: Jun 24, 2012
  2. jcsd
  3. Jun 24, 2012 #2
    Your integration of dv is wrong.
     
  4. Jun 24, 2012 #3

    SammyS

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    Staff Emeritus
    Science Advisor
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    Gold Member

    What is it that you're trying to integrate? What you wrote is ambiguous.

    [itex]\displaystyle \int_0^1 \frac{y}{e^{2y}}\,dy\,,[/itex]

    or

    [itex]\displaystyle \int_0^1 y\,e^{2y}\,dy\,,[/itex]

    or something else?
     
  5. Jun 24, 2012 #4

    Mark44

    Staff: Mentor

    In addition to what 1MileCrash said, you aren't doing integration by parts correctly. If you integral is ##\int ye^{-2y}dy##
    then you can't have u = e2y and dv = 1/(e2y)* dy.
    Whatever you choose for u and dv, they have to multiply to make your original integrand. In your case u*dv = e2y/e2y * dy = 1 dy, not ye-2ydy.
     
  6. Jun 24, 2012 #5
    So your choice of u and dv should be very clear. As for which one is u and which one is dv - weigh which one will result in a simpler function when differentiated, and try that for u.
     
  7. Jun 24, 2012 #6
    It is y divided by e^2y dy
     
  8. Jun 24, 2012 #7

    Mark44

    Staff: Mentor

    That's what I understood it to be.

    You should write this not as a fraction, but as ye-2ydy.
     
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