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Math Help - Integrate (sinx-5cosx)/(sinx+cosx)

  1. #1
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    Integrate (sinx-5cosx)/(sinx+cosx)

    I'm not sure how to integrate this function. I tried converting the numerator into a(sinx+cosx)+b(cosx-sinx), but I couldn't find anything that worked. It didn't really simplify the integrand.
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    Re: Integrate (sinx-5cosx)/(sinx+cosx)

    Hello, quanta123!

    I found a way, but it's pretty elaborate.
    I'm sure there is a simpler way.

    I used these identities:
    . . . . . \sin^2x \:=\:\frac{1-\cos2x}{2}
    . . . . . \cos^2x \:=\:\frac{1+\cos2x}{2}
    n 2\sin x\cos x \:=\:\sin2x

    \cos^2\!x - \sin^2\!x \:=\:\cos2x


    \text{Integrate: }\:\int\frac{\sin x - 5\cos x}{\sin x - \cos x}\,dx

    Multiply by \frac{\sin x + \cos x}{\sin x + \cos x}:

    . . \frac{\sin x - 5\cos x}{\sin x - \cos x}\cdot\frac{\sin x+\cos x}{\sin x+\cos x} \;=\; \frac{\sin^2\!x + \sin x\cos x - 5\sin x\cos x - 5\cos^2\!x}{\sin^2\!x-\cos^2\!x}

    . . =\;\frac{\sin^2\!x - 4\sin x\cos x - 5\cos^2\!x}{-(\cos^2\!x - \sin^2\!x)} \;=\;\frac{\frac{1-\cos2x}{2} - 2\sin2x - 5\left(\frac{1+\cos2x}{2}\right)}{-\cos2x}

    . . =\;\frac{1-\cos2x - 4\sin2x -5-5\cos2x}{-2\cos2x} \;=\;\frac{-4-4\sin2x - 6\cos2x}{-2\cos2x}

    . . =\;\frac{2 + 2\sin2x + 3\cos2x}{\cos2x} \;=\; \frac{2}{\cos2x} + \frac{2\sin2x}{\cos2x} + \frac{3\cos2x}{\cos2x}

    . . =\;2\sec2x + 2\tan2x + 3


    Got it?
    Thanks from sbhatnagar
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    Re: Integrate (sinx-5cosx)/(sinx+cosx)

    Integrate (sinx-5cosx)/(sinx+cosx)-good-integral.png
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  4. #4
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    Re: Integrate (sinx-5cosx)/(sinx+cosx)

    I think there's a simplest way...

    \displaystyle \int \dfrac{\sin x -5\cos x}{\sin x -\cos x}dx=\int \dfrac{\sin x -cos x -4\cos x}{\sin x - \cos x}dx=\int dx-4\int \dfrac{\cos x}{\sin x -\cos x}dx

    and for the last one integral, using t=\tan \frac{x}{2} will be using simple algebra with a rectangle triangle with t,1,\sqrt{t^2+1} sides
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