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Math Help - Powers of sines and cosines

  1. #1
    Newbie Elvis's Avatar
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    Powers of sines and cosines

    Say we want to find \int\sin^m{x}\cos^n{x}\;{dx}. Then we would make use of the identity \cos^2{x}+\sin^2{x}= 1, if either m or n is odd, and \cos^2x = \dfrac{1+\cos{2x}}{2} and \sin^2x = \dfrac{1-\cos{2x}}{2}, if m and n are both even. But isn't this impractical when m and n are large? When want to find, for example, \int\sin^{78}{x}\cos^{36}{x}\;{dx}? Or am I missing some trick?
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  2. #2
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    Quote Originally Posted by Elvis View Post
    Say we want to find \int\sin^m{x}\cos^n{x}\;{dx}. Then we would make use of the identity \cos^2{x}+\sin^2{x}= 1, if either m or n is odd, and \cos^2x = \dfrac{1+\cos{2x}}{2} and \sin^2x = \dfrac{1-\cos{2x}}{2}, if m and n are both even. But isn't this impractical when m and n are large? When want to find, for example, \int\sin^{78}{x}\cos^{36}{x}\;{dx}? Or am I missing some trick?
    You can create reduction formulae using integration by parts.

    These reduction formulae are:

    \int{\sin^m{x}\cos^n{x}\,dx} = \frac{\sin^{m + 1}{x}\cos^{n - 1}{x}}{m + n} + \frac{n - 1}{m + n}\int{\sin^m{x}\cos^{n - 2}{x}\,dx}

    OR

    \int{\sin^m{x}\cos^n{x}\,dx} = -\frac{\sin^{m - 1}{x}\cos^{n + 1}{x}}{m + n} + \frac{m - 1}{m + n}\int{\sin^{m - 2}{x}\cos^n{x}\,dx}.
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