The value of $\int_{-\pi / 6}^{\pi / 6}\left(\frac{\pi+4 x^{11}}{1-\sin (|x|+\pi / 6)}\right) d x$ is equal to

JEE Main · 2026 · 21 Jan 2026 · Shift 1 · Definite Integrals

The value of

\int_{- π /6}^{π /6} (\frac{π + 4 x ^{11}}{1 - s i n ( ∣ x ∣ + π /6 )}) d x

is equal to

Answer: (B)

The term

4 x^{11}

is odd while the denominator depends only on

∣ x ∣

, so that part integrates to zero over the symmetric interval. By evenness,

\int_{- π /6}^{π /6} \frac{π d x}{1 - s i n ( ∣ x ∣ + π /6 )} = 2 π \int_{0}^{π /6} \frac{d x}{1 - s i n ( x + π /6 )}

. Substituting

t = x + \frac{π}{6}

and using

\frac{1}{1 - s i n t} = sec^{2} t + sec t tan t

, this becomes

2 π [tan t + sec t]_{π /6}^{π /3} = 2 π ((3 + 2) - 3) = 4 π

JEE Main · 2026 · 21 Jan 2026 · Shift 1 · Single correct option

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