The value of the integral 5 24 24 tan2x is: [2026]

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Solution

Q.
The value of the integral $\int_{\frac{π}{24}}^{\frac{5 π}{24}} \frac{d x}{1 + \sqrt[3]{\tan 2 x}}$ is: [2026]

1 $\frac{π}{3}$

2 $\frac{π}{18}$

3 $\frac{π}{12}$

4 $\frac{π}{6}$

Ans.
(3)

$I = \int_{\frac{π}{24}}^{\frac{5 π}{24}} \frac{d x}{1 + \sqrt[3]{\tan 2 x}} . . . (1)$

Apply King

$I = \int_{\frac{π}{24}}^{\frac{5 π}{24}} \frac{d x}{1 + \sqrt[3]{\tan 2 (\frac{π}{4} - x)}}$

$I = \int_{\frac{π}{24}}^{\frac{5 π}{24}} \frac{d x}{1 + \sqrt[3]{\cot 2 x}} . . . (2)$

Add (1) + (2) $\Rightarrow 2 I = \int_{\frac{π}{24}}^{\frac{5 π}{24}} (1) d x$

$I = \frac{1}{2} (\frac{π}{6}) = \frac{π}{12}$

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