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Thread: Integration by Parts Help Please :( Tough Question

  1. #1
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    Unhappy Integration by Parts Help Please :( Tough Question

    a/ Use intergration by parts to express:



    in terms of I(n - 2)

    b/ Hence, show that

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  2. #2
    Junior Member
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    It's a bit tricky, but here we go:

    $\displaystyle I(n)= \int \sin^{n}(x) dx = \int \sin^{n-1}(x) \sin(x)dx$

    Now we use integration by parts with $\displaystyle u=\sin^{n-1}(x), dv=\sin(x)dx $ and $\displaystyle du=(n-1)\sin^{n-2}(x)\cos(x)dx, v=-cos(x)dx$. So we get

    $\displaystyle I(n)=-\sin^{n-1}(x)\cos(x)+(n-1)\int\sin^{n-2}(x)\cos^2(x)dx $

    Using the identity $\displaystyle \sin^2(x)+\cos^2(x)=1$, we get

    $\displaystyle I(n)=-\sin^{n-1}(x)\cos(x)+(n-1)\int\sin^{n-2}(x)(1-\sin^2(x))dx$

    The integral on the RHS we recognize as $\displaystyle I(n-2)-I(n)$ so we get

    $\displaystyle I(n) = -\sin^{n-1}(x)\cos(x) + (n-1)\left(I(n-2)-I(n)\right)$

    Well the rest is very easy, just solve for $\displaystyle I(n) $ and you should have (a).
    Putting it to the test for your given integral, we need $\displaystyle I(n-2)$ expressed in $\displaystyle I(n)$. Some basic manipulation should get you

    $\displaystyle I(n-2) = \frac{n}{n-1}I(n) + \frac{\sin^{n-1}(x)\cos(x)}{n-1}$

    Now your asked integral is basically $\displaystyle I(-4)=I(-2-2)$, with boundaries filled in. You should be able to do the rest on your own. I can confirm that the answer is indeed $\displaystyle \frac{4}{3}$.
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