Question

Sol. [Key Concept: The integrand is an exact derivative — the antiderivative is $\frac{cot\ x}{\cos^{4}x}$; evaluate it at $\pi/6$ and $\pi/3$.]

$$= \int_{\pi/6}^{\pi/3}\mspace{2mu}\frac{4}{\cos^{4}x}dx - \int_{\pi/6}^{\pi/3}\mspace{2mu}\frac{{cosec}^{2}x}{\cos^{4}x}dx$$

$$\begin{matrix} & \ = \int_{\pi/6}^{\pi/3}\mspace{2mu}\mspace{2mu}\frac{4}{\cos^{4}x}dx - \left\lbrack - \frac{cotx}{\cos^{4}x} - \int_{\pi/6}^{\pi/3}\mspace{2mu}\mspace{2mu}( - cotx)\frac{- 4}{\cos^{5}x}( - sinx)dx \right\rbrack \\ & \ = \left. \ \frac{cotx}{\cos^{4}x} \right|_{\frac{\pi}{6}}^{\frac{\pi}{3}} \\ & \ = \frac{1/\sqrt{3}}{(1/2)^{4}} - \frac{\sqrt{3}}{\left( \frac{\sqrt{3}}{2} \right)^{4}} \end{matrix}$$