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Integration by Parts

  1. Apr 12, 2007 #1
    Hi,

    [tex] J(m,n) = \int_0^{\frac{\pi}{2}} \cos^m \theta \sin^n \theta d\theta [/tex]


    First of all I had to evaluate the following ( I don't know what the correct answers are but here are my calculations:

    [tex] J(0,0) = [\theta]_0^{\frac{{\pi}{2}}}=\frac{\pi}{2} [/tex]

    [tex] J(0,1) = [-\cos \theta]_0^{\frac{{\pi}{2}}}= 1 [/tex]

    [tex] J(1,0) = [ \sin\theta]_0^{\frac{{\pi}{2}}}= 1 [/tex]

    [tex] J(1,1) = [\frac{-\cos 2\theta}{4}]_0^{\frac{{\pi}{2}}}= \frac{1}{2} [/tex]

    [tex] J(m,1) = [-\frac{\cos^{m+1} \theta}{m+1}]_0^{\frac{{\pi}{2}}}= \frac{1}{m+1} [/tex]

    [tex] J(1,n) = [\frac{\sin^{n+1} \theta}{n+1}]_0^{\frac{{\pi}{2}}}= \frac {1}{n+1} [/tex]

    Next I am supposed to use integration by parts to prove that for m and n > 1

    [tex] J(m,n) = \frac{m-1}{m+n} J(m-2,n) [/tex]

    and

    [tex] J(m,n) = \frac{n-1}{m+n} J(m,n-2) [/tex]

    When I carried out integration by parts I got the following:

    taking

    [tex] u = \sin^{n-1} \theta [/tex]
    [tex] u' = (n-1) \sin^{n-2}\theta \cos \theta [/tex]
    [tex] v' = \cos^m \theta \sin \theta [/tex]
    [tex] v = -\frac{cos^{m+1}\theta}{m+1} [/tex]

    [tex] \frac{-\sin^{n-1} \theta \cos^{m+1}\theta}{m+1} + \int_0^{\frac{\pi}{2}} \frac{n-1}{m+1} \sin^{n-2} \theta \cos^{m+2} \theta d\theta [/tex]

    the first term on the RHS equates to zero but the 2nd term is not correct: the denominator and the power to which cos is raised is wrong but I'm not sure how to fix it
     
    Last edited: Apr 12, 2007
  2. jcsd
  3. Apr 13, 2007 #2

    VietDao29

    User Avatar
    Homework Helper

    I dunno why, but your LaTeX part is a bit messy for me. :frown:
    You are correct to the last part. So, you have shown that:
    [tex]\int_0 ^ \frac{\pi}{2} \cos ^ m \theta \sin ^ n \theta d( \theta ) = \frac{n - 1}{m + 1} \int_0 ^ \frac{\pi}{2} \cos ^ {\fbox{m + 2}} \theta \sin ^ {\fbox{n - 2}} \theta d( \theta )[/tex]

    Note that cos function on the RHS is to the power of m + 2, and you want to prove that:

    [tex]\int_0 ^ \frac{\pi}{2} \cos ^ m \theta \sin ^ n \theta d( \theta ) = \frac{n - 1}{m + n} \int_0 ^ \frac{\pi}{2} \cos ^ {\fbox{m}} \theta \sin ^ {\fbox{n - 2}} \theta d( \theta )[/tex].

    So what you should do is to split cosm + 2(x) to cosm(x)cos2(x) = cosm(1 - sin2(x)), like this:
    [tex]RHS = \frac{n - 1}{m + 1} \int_0 ^ \frac{\pi}{2} \cos ^ {m} \theta (1 - \sin ^ 2 \theta) \sin ^ {n - 2} \theta d( \theta ) = \frac{n - 1}{m + 1} \left( \int_0 ^ \frac{\pi}{2} \cos ^ {m} \theta \sin ^ {n - 2} \theta d( \theta ) - \int_0 ^ \frac{\pi}{2} \cos ^ {m} \theta \sin ^ {n} \theta d( \theta ) \right)[/tex].

    So we have:

    [tex]J(m, n) = \frac{n - 1}{m + 1} \left( J(m, n - 2) - J(m, n) \right)[/tex]
    Can you go from here? :)

    --------

    The other can be done almost the same, except for the first step:
    Instead of choosing
    u = sinn - 1(x)
    and dv = cosm(x) sin(x) dx, we choose:
    u = cosm - 1(x)
    and dv = sinn(x) cos(x) dx.

    Can you complete the two problems? :smile:
     
    Last edited: Apr 13, 2007
  4. Apr 15, 2007 #3
    Hey VietDao29!

    Thanks for your help on this.

    I assume that the rest of the answer rely's on my using the reduction formula for trigonometric integrals?

    I've had a quick scoot on the web and there seems to be two main cases m+n odd or m+n even. Which uses two different formula (which does make me a little anxious as there is only one formula that we are trying to prove.)

    Anyway I've included a link to the worksheet I am working from.

    http://www.math.ualberta.ca/~apotapov/MATH115/trinth.pdf

    Is this the correct method to solve the problem?
     
  5. Apr 17, 2007 #4
    Hi VietDao,

    I managed to figure it out.

    Your hint brought me alot closer to the solution than I initially realised!

    Thanks again :smile:
     
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