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Substitution method with Integration by Parts?

  1. Jan 17, 2012 #1
    Substitution method with Integration by Parts???

    1. The problem statement, all variables and given/known data

    Evaluate the integral...
    ∫x^3[e^(-x^2)]dx


    2. Relevant equations

    ∫udv=uv-∫vdu

    3. The attempt at a solution
    I first tried using integration by parts setting u and dv equal to anything and everything. This seemed to make it more complicated so I decided to use the substitution method setting y=-x^2 and dy=-2xdx and finally -1/2dy=xdx to give me
    1/2∫ye^ydy
    From here i used integration by parts with u=y, du=dy, v=e^y and dv=e^y dy to get
    1/2(ye^y-∫e^y dy) = 1/2(ye^y-e^y)
    and then
    1/2[-x^2e^(-x^2)-e^(-x^2)]
    This problem got really confusing with all the variables and I just want to make sure that I'm not way off base with my methods and solution.
    Thanks in advance for any input!
     
  2. jcsd
  3. Jan 17, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Substitution method with Integration by Parts???

    Looks right to me.
     
  4. Jan 17, 2012 #3
    Re: Substitution method with Integration by Parts???

    That's a relief! I spent a loooonnnggg time working this problem. Thanks!
     
  5. Jan 17, 2012 #4

    Mark44

    Staff: Mentor

    Re: Substitution method with Integration by Parts???

    Once you have gotten an antiderivative, it's a good practice to check it by differentiating it. If your work is correct, the derivative of your answer should be the original integrand.
     
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