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Homework Help: Integration by parts

  1. Sep 5, 2009 #1
    1. The problem statement, all variables and given/known data


    Use integration by parts to verify the formula:
    [tex]\int x^{n} sin x dx = -x^{n} cos x + n\int x^{n-1} cos x dx[/tex]

    2. Relevant equations

    3. The attempt at a solution

    For the first one, I attached the picture of my work on paper, as it would take me forever to type out in latex code, I think. For the second one:

    [tex]u = sin x[/tex]
    [tex]du = cos x[/tex]
    [tex]dV = x^{n}[/tex]
    [tex]V = \frac{x^{n+1}}{n+1}[/tex]

    [tex]\int x^{n} sin x dx = -x^{n} cos x + n\int x^{n-1} cos x dx[/tex] =
    [tex]sin x (\frac{x^{n+1}}{n+1}) - \int \frac{x^{n+1}}{n+1} cos x dx[/tex]

    That doesn't really look like the formula to me. Am I supposed to use an identity of some sorts?

    Attached Files:

  2. jcsd
  3. Sep 5, 2009 #2
    I can't see the work for the first one, so I can't tell if that's right or not.

    For the second one, you differentiated [tex]\sin{x}[/tex] to get [tex]\cos{x}[/tex]and integrated [tex]x^n[/tex] to get [tex]\frac{x^{n+1}}{n+1}[/tex], is that right? However, since the formula you're supposed to end up with has an [tex]x^{n-1}[/tex], I would differentiate the [tex]x^n[/tex] and integrate the [tex]\sin{x}[/tex] and see what you get.
  4. Sep 5, 2009 #3
    mathie.girl is right. A good thing to remember when doing integration by parts is that you let u be the term such that when you differentiate it, du is "simpler" than u. For example, you let u=sinx so that du=cosx. That really doesn't do any simplifying. That means you'll let dv be the term that doesn't really simplify when taking the derivative.
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