# 14 $\\int_{\\frac{1}{6}}^{\\frac{1}{2}}\\csc(\\pi a)\\cot(\\pi a)da$

Explanation:

Step1: Let (u = \pi A)

Differentiate (u) with respect to (A): (du=\pi dA), so (dA=\frac{du}{\pi}). When (A = \frac{1}{6}), (u=\frac{\pi}{6}); when (A=\frac{1}{2}), (u=\frac{\pi}{2}). The integral becomes (\frac{1}{\pi}\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\csc(u)\cot(u)du).

Step2: Integrate (\csc(u)\cot(u))

The integral of (\csc(u)\cot(u)) is (-\csc(u)). So (\frac{1}{\pi}\left[-\csc(u)\right]_{\frac{\pi}{6}}^{\frac{\pi}{2}}).

Step3: Evaluate the definite integral

[ \begin{align*} \frac{1}{\pi}\left(-\csc\left(\frac{\pi}{2}\right)+\csc\left(\frac{\pi}{6}\right)\right)&=\frac{1}{\pi}(- 1 + 2)\ &=\frac{1}{\pi} \end{align*} ]

Answer:

(\frac{1}{\pi})