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Math Help - A complicated integral

  1. #1
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    A complicated integral

    This is an integral for finding length of a segment of a parametric graph.
    \int \sqrt{(\frac{1}{2\sin .5t \cos .5t} - \sin t)^2 + (\cos t)^2}

    I simplified it down to this
    \int |\cot t|

    How do you go about solving this for the interval from \frac{\pi}{4} to \frac{3\pi}{4}?
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by cubs3205 View Post
    I simplified it down to this
    \int |\cot t|
    Note that  \int |\cot(t)|dt = \frac{|\cot(t)|}{\cot(t)}\ln(\sin(t))+C

    This is because  \int |f(x)| dx = \frac{|f(x)|}{f(x)} \cdot \int f(x) dx
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  3. #3
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    \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} |\cot{t}| \, dt = 2\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot{t} \, dt = 2\left[\ln|\sin{t}|\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} = \ln(2)
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