1. ## A complicated integral

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

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

How do you go about solving this for the interval from $\displaystyle \frac{\pi}{4} to \frac{3\pi}{4}$?

2. Originally Posted by cubs3205
I simplified it down to this
$\displaystyle \int |\cot t|$
Note that $\displaystyle \int |\cot(t)|dt = \frac{|\cot(t)|}{\cot(t)}\ln(\sin(t))+C$

This is because $\displaystyle \int |f(x)| dx = \frac{|f(x)|}{f(x)} \cdot \int f(x) dx$

3. $\displaystyle \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} |\cot{t}| \, dt = 2\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot{t} \, dt =$$\displaystyle 2\left[\ln|\sin{t}|\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} = \ln(2)$