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Homework Help: Integration by parts question

  1. Jan 3, 2015 #1


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    The problem statement, all variables and given/known data
    $$ \int x^{3}cos(x^{2})dx$$

    The attempt at a solution
    OK, so I am aware that there is a way in which to do this problem where you do a substitution (let $$u=x^{2}$$ to do a substitution before you integrate by parts), and I was able to get the answer right using this method. The thing is, I tried it a different way first, and after triple checking it, I feel like I should have gotten it right. It doesn't matter what I choose to let u and dv equal, right? I should get the right answer no matter what I choose (assuming I did things correctly, of course).
    Here's my work:


    $$\frac{x^{4}cos(x^{2})}{4}+\frac{1}{2}\int x*sin(x^{2})dx$$


    $$\frac{x^{4}cos(x^{2})}{4}+\frac{1}{2}(\frac{x^{2}sin(x^{2})}{2}-\int x^{3}cos(x^{2})dx)$$

    Since this last integral is the same as the one we started with, we can now say that
    Multiplying both sides by two over three...

    Thanks for your time and any help you can offer me.
  2. jcsd
  3. Jan 3, 2015 #2


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    Staff Emeritus
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    So, are you asking why your result isn't correct?

    ##\displaystyle v\,du = (-1/2)x^5\sin(x^2)\,dx\ \text{ not } (-1/2)x\sin(x^2)\,dx\ .##
  4. Jan 3, 2015 #3


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    I knew it was something stupid. No idea where I got x from. Thanks Sammy.
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