The value of the integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\left(\frac{4-\operatorname{cosec}^{2} x}{\cos ^{4} x}\right) d x$ is :

JEE Main · 2026 · 05 Apr 2026 · Shift 1 · Definite Integrals

The value of the integral

\int_{\frac{π}{6}}^{\frac{π}{3}} (\frac{4 - cosec ^{2} x}{c o s ^{4} x}) d x

is :

Answer: (C)

Since

\frac{d}{d x} (\frac{cot x}{cos ^{4} x}) = - \frac{csc ^{2} x}{cos ^{4} x} + \frac{4 cot x tan x}{cos ^{4} x} = \frac{4 - csc ^{2} x}{cos ^{4} x}

, the function

\frac{cot x}{cos ^{4} x}

is an antiderivative of the integrand. Evaluating,

[\frac{cot x}{cos ^{4} x}]_{π /6}^{π /3} = \frac{1/ 3}{( 1/2 ) ^{4}} - \frac{3}{( 3 /2 ) ^{4}} = \frac{16}{3} - \frac{16 3}{9} = \frac{48 3 - 16 3}{9} = \frac{32 3}{9}

. Hence the value of the integral is

\frac{32}{3 3}

JEE Main · 2026 · 05 Apr 2026 · Shift 1 · Single correct option

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