QJEEM19Ma4ae8: JEE Main 2019 Mathematics - Class 12 JEEM19Ma4ae8 Solution: The integral is equal to :

Question

Verified Solution • 2025–26 Session moderate

The integral

\int\limits_{\pi /6}^{\pi /3} {{{\sec }^{2/3}}} x\cos e{c^{4/3}}xdx

is equal to :

Accepted Answer

\int\limits_{\pi /6}^{\pi /3} {{{\sec }^{2/3}}x\,{{{\mathop{\rm cosec}\nolimits} }^{4/3}}x\,dx}

= \int {{{{{\sec }^2}x} \over {{{\tan }^{4/3}}x}}dx}

Let tan x = t, sec² x dx = dt

= \int {{{dt} \over {{t^{4/3}}}}}

I = - 3\left( {{t^{ - 1/3}}} \right)

\Rightarrow \left( { - 3{{\left( {\tan x} \right)}^{{{ - 1} \over 3}}}} \right)_{\pi /6}^{\pi /3}

\Rightarrow 3\left( {{3^{{1 \over 3}}} - {1 \over {{3^{{1 \over 6}}}}}} \right)

= 3^7/6 - 3^5/6

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