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Math Help - reduction formula proof

  1. #1
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    reduction formula proof

    How would I prove the following reduction formula.

    ∫cos^n xdx=1/n (cos^n-1 x)sinx+(n-1/n) ∫cos^n-2 xdx?

    I know there are plenty of sites that explain this but I want an answer from a trusted source.
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  2. #2
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    Quote Originally Posted by gammaman View Post
    How would I prove the following reduction formula.

    ∫cos^n xdx=1/n (cos^n-1 x)sinx+(n-1/n) ∫cos^n-2 xdx?

    I know there are plenty of sites that explain this but I want an answer from a trusted source.
    Hi

    Let I_n = \int \cos^nx \:dx

    I_n = \int \cos^2x \:cos^{n-2}x \:dx

    I_n = \int (1-\sin^2x) \:cos^{n-2}x \:dx

    I_n = \int cos^{n-2}x \:dx - \int \sin^2x \:cos^{n-2}x \:dx

    I_n = \int cos^{n-2}x \:dx - \int \sin x \:\sin x\:cos^{n-2}x \:dx

    By parts with u = \sin x and dv = \sin x\:cos^{n-2}x \:dx = -\frac{1}{n-1}\:d\left(cos^{n-1}x\right)

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

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

    \left(1 + \frac{1}{n-1}\right)\: I_n = \int cos^{n-2}x \:dx + \frac{1}{n-1}\:\sin x \: cos^{n-1}x

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

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