If $n$ is an odd natural number, then $\mathop \smallint \limits_{ - \pi /6}^{\pi /6} \,\,\frac{{\pi \,\, + \,\,4{x^n}}}{{1 - \sin \,\,\left( {pq\,\, + \,\,\frac{\pi }{6}} \right)}}\,\,dx = $

$2\pi \,\, + \,\,\frac{1}{{\sqrt 3 }}$

$4\pi \,\, + \,\,\sqrt 3 \,\, - \,\,\frac{1}{{\sqrt 3 }}$

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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Application +4 | -1

n

is an odd natural number, then

\int \limits_{ - \pi /6}^{\pi /6} \,\,\frac{{\pi \,\, + \,\,4{x^n}}}{{1 - \sin \,\,\left( {pq\,\, + \,\,\frac{\pi }{6}} \right)}}\,\,dx =

View answer and explanation

Correct option: A

4\,\,\pi

Option A (

4\pi

) is correct. The integral simplifies due to the odd function and trigonometric identities, resulting in the value

4\pi

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