Ta có:
$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}}\cot^2xdx$
$=\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}}\dfrac{\cos^2xdx}{\sin^2x}$
$=\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left(\dfrac{1}{\sin^2x}-1\right)dx$
$=(-\cot x-x)\left|\begin{array}{l}\dfrac{\pi}{2}\\\dfrac{\pi}{4}\end{array}\right.=1-\dfrac{\pi}{4}$