## Calculus: Early Transcendentals 8th Edition

$\displaystyle\int\limits_{\pi/6}^{\pi/2}\csc t\cot tdt=1$
$\displaystyle\int\limits_{\pi/6}^{\pi/2}\csc t\cot tdt$ We know that $\dfrac{d}{dx}\csc t=-\cot t\csc t$. So we can integrate this expression directly and then apply the second part of the fundamental theorem of algebra: $\displaystyle\int\limits_{\pi/6}^{\pi/2}\csc t\cot tdt=-\csc t\Big|_{\pi/6}^{\pi/2}=-(\csc\dfrac{\pi}{2}-\csc\dfrac{\pi}{6})=-(1-2)=1$