The integral $\int_{\pi / 6}^{\pi / 3} \sec ^{2 / 3} x \operatorname{cosec}^{4 / 3} x d x$ is equal to

JEE Main · 2019 · 10 Apr 2019 · Shift 2 · Definite Integrals

The integral

\int_{π /6}^{π /3} sec^{2/3} x cosec^{4/3} x d x

is equal to

Answer: (C)

Write the integrand as

\frac{s e c ^{2} x}{t a n ^{4/3} x}

and substitute

u = tan x

d u = sec^{2} x d x

, turning it into

\int u^{- 4/3} d u = - 3 u^{- 1/3} = - \frac{3}{t a n ^{1/3} x}

. Evaluating from

\frac{π}{6}

\frac{π}{3}

, where

tan \frac{π}{3} = 3

and

tan \frac{π}{6} = \frac{1}{3}

, gives

- 3 (3^{- 1/6} - 3^{1/6}) = 3^{7/6} - 3^{5/6}

JEE Main · 2019 · 10 Apr 2019 · Shift 2 · Single correct option

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