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Math Help - How do I integrate (sinx)^6 ?

  1. #1
    s3a
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    How do I integrate (sinx)^6 ?

    I could integrate (sinx)^4 by doing (sin^2 x)^2 for example and then using the half angle formula (1 - cos(2x))/2 but I can't do that for (sinx)^6 and Wolfram tells me to use a reduction formula which is not something I am supposed to know yet.

    So can someone please help me to do this integral? It's not the actual problem I am doing but is part of the problem that I think I am stuck on. (My real homework problem is: the Integral of (sinx)^4 * (cosx)^2 dx.

    Any help would be greatly appreciated!
    Thanks in advance!
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    Quote Originally Posted by s3a View Post
    I could integrate (sinx)^4 by doing (sin^2 x)^2 for example and then using the half angle formula (1 - cos(2x))/2 but I can't do that for (sinx)^6 and Wolfram tells me to use a reduction formula which is not something I am supposed to know yet.

    So can someone please help me to do this integral? It's not the actual problem I am doing but is part of the problem that I think I am stuck on. (My real homework problem is: the Integral of (sinx)^4 * (cosx)^2 dx.

    Any help would be greatly appreciated!
    Thanks in advance!
    repeated use of the double angle power reduction identity ...

    \int \sin^2{x} \cdot \sin^2{x} \cdot \cos^2{x} \, dx<br />

    \int \left[\frac{1-\cos(2x)}{2}\right] \cdot \left[\frac{1-\cos(2x)}{2}\right] \cdot \left[\frac{1+\cos(2x)}{2}\right] \, dx

    \int \left[\frac{1-\cos(2x)}{2}\right] \cdot \left[\frac{1-\cos^2(2x)}{4}\right] \, dx

    \int \left[\frac{1-\cos(2x)}{2}\right] \cdot \left[\frac{\sin^2(2x)}{4}\right] \, dx

    \frac{1}{8}\int \sin^2(2x) - \sin^2(2x) \cdot \cos(2x) \, dx

    \frac{1}{8} \int \sin^2(2x) \, dx - \frac{1}{8}\int \sin^2(2x) \cdot \cos(2x) \, dx

    \frac{1}{16} \int 1 - \cos(4x) \, dx - \frac{1}{8} \int \sin^2(2x) \cdot \cos(2x) \, dx

    \frac{1}{16}\left[x - \frac{\sin(4x)}{4}\right] - \frac{1}{16} \cdot \frac{\sin^3(2x)}{3} + C
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