The value of the definite integral `int_(pi//24)^(5pi//24)(dx)/(1+root(3)(tan2x))` is :

Options

A. `(pi)/(3)`
B. `(pi)/(6)`
C. `(pi)/(12)`
D. `(pi)/(18)`

Answer

To solve the integral

\[ I = \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan(2x)}} \]

we can use a substitution technique that involves replacing \( x \) with \( \frac{\pi}{4} - x \).

### Step 1: Substitute \( x = \frac{\pi}{4} - x \)

First, we calculate the new limits of integration:
- When \( x = \frac{\pi}{24} \):
\[ \frac{\pi}{4} - \frac{\pi}{24} = \frac{6\pi}{24} - \frac{\pi}{24} = \frac{5\pi}{24} \]
- When \( x = \frac{5\pi}{24} \):
\[ \frac{\pi}{4} - \frac{5\pi}{24} = \frac{6\pi}{24} - \frac{5\pi}{24} = \frac{\pi}{24} \]

Thus, the integral becomes:
\[ I = \int_{\frac{5\pi}{24}}^{\frac{\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan(2(\frac{\pi}{4} - x))}} \]

### Step 2: Simplify the integrand

Using the identity \( \tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta) \):
\[ \tan(2(\frac{\pi}{4} - x)) = \tan\left(\frac{\pi}{2} - 2x\right) = \cot(2x) \]

Thus, we can rewrite the integral:
\[ I = \int_{\frac{5\pi}{24}}^{\frac{\pi}{24}} \frac{dx}{1 + \sqrt[3]{\cot(2x)}} \]

### Step 3: Change the limits of integration

Reversing the limits of integration gives:
\[ I = -\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\cot(2x)}} \]

### Step 4: Combine the two expressions for \( I \)

Now, we have two expressions for \( I \):
1.
\[ I = \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan(2x)}} \]
2.
\[ I = -\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\cot(2x)}} \]

Adding these two equations gives:
\[ 2I = \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \left( \frac{1}{1 + \sqrt[3]{\tan(2x)}} + \frac{1}{1 + \sqrt[3]{\cot(2x)}} \right) dx \]

### Step 5: Simplify the integrand

Notice that:
\[ \frac{1}{1 + \sqrt[3]{\tan(2x)}} + \frac{1}{1 + \sqrt[3]{\cot(2x)}} = 1 \]

Thus, we have:
\[ 2I = \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} 1 \, dx \]

### Step 6: Calculate the integral

Now, we can compute the integral:
\[ 2I = \left[ x \right]_{\frac{\pi}{24}}^{\frac{5\pi}{24}} = \frac{5\pi}{24} - \frac{\pi}{24} = \frac{4\pi}{24} = \frac{\pi}{6} \]

### Step 7: Solve for \( I \)

Now, divide both sides by 2:
\[ I = \frac{\pi}{12} \]

### Final Result

Thus, the value of the definite integral is:
\[ \boxed{\frac{\pi}{12}} \]