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$Old$ January 14th, 2009, 03:02 PM

cubs3205 $cubs3205 is offline$

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$Default$ 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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$Old$ January 14th, 2009, 03:15 PM

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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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$Old$ January 14th, 2009, 03:20 PM

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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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