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Math Help - how to split this integral..

  1. #1
    MHF Contributor
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    how to split this integral..

    <br />
\int_{0}^{\frac{\pi}{2}}\left|\frac{sin(\frac{3\pi  }{2}+x)}{(3+\cos^2 x)(3-\sin x)}\right| dx<br />
    the denominator is positive the numerator needs to be splited

    sinx>0 when 0<x<\pi

    0<\frac{3\pi}{2}+x<\pi
    -\frac{3\pi}{2}<x<-\frac{\pi}{2}

    what now ??
    how to split the integral and cancel the absolute value??
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  2. #2
    Member Mentia's Avatar
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    So the argument of the sine in the numerator goes from \frac{3 \pi}{2} + 0 = \frac{3 \pi}{2} to \frac{3 \pi}{2} + \frac{\pi}{2} = 2\pi

    Sine(x) in this domain is negative. So you simply require a negative sign in front of the integral and remove the absolute value signs:

    \int_{0}^{\frac{\pi}{2}}-\frac{sin(\frac{3\pi}{2}+x)}{(3+\cos^2 x)(3-\sin x)} dx<br />
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  3. #3
    MHF Contributor
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    thanks
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