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Math Help - need help with reduction formula

  1. #1
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    need help with reduction formula












    for the last part, if I tranpose the last integral to the left side, the constant, should it still be (n-1)? why does it changes to n instead?
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  2. #2
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    Quote Originally Posted by justin016 View Post
    for the last part, if I tranpose the last integral to the left side, the constant, should it still be (n-1)? why does it changes to n instead?
    Moving the rightmost integral in your penultimate line to the left-hand side, you have

    \int \sin^nx\,dx+(n-1)\int\sin^nx\,dx=-\sin^{n-1}x\cos x+(n-1)\int\sin^{n-2}x\,dx

    \implies(1+n-1)\int\sin^n x\,dx = -\sin^{n-1}x\cos x + (n-1)\int\sin^{n-2}x\,dx

    \implies n\int\sin^nx\,dx = -\sin^{n-1}x\cos x + (n-1)\int\sin^{n-2}x\,dx

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