The value of the integral $\int_{\frac{\pi}{24}}^{\frac{5 \pi}{24}} \frac{\mathrm{dx}}{1+\sqrt[3]{\tan 2 \mathrm{x}}}$ is :

JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Definite Integrals

The value of the integral

\int_{\frac{π}{24}}^{\frac{5 π}{24}} \frac{dx}{1 + 3 t a n 2 x}

is :

Answer: (A)

Let

I = \int_{\frac{π}{24}}^{\frac{5 π}{24}} \frac{d x}{1 + 3 t a n 2 x}

.

Using the King property with

x \to \frac{π}{4} - x

tan 2 (\frac{π}{4} - x) = cot 2 x

.

So

I = \int_{\frac{π}{24}}^{\frac{5 π}{24}} \frac{d x}{1 + 3 c o t 2 x}

.

Adding the two expressions:

2 I = \int_{\frac{π}{24}}^{\frac{5 π}{24}} (\frac{1}{1 + 3 t a n 2 x} + \frac{1}{1 + 3 c o t 2 x}) d x = \int_{\frac{π}{24}}^{\frac{5 π}{24}} 1 d x

.

Thus

2 I = \frac{5 π}{24} - \frac{π}{24} = \frac{π}{6}

, so

I = \frac{π}{12}

JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Single correct option

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