Evaluate : `int_((pi)/(5))^((3pi)/(10))(dx)/(tan^(n) x +1)`

Answer

To evaluate the integral

\[ I = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{dx}{\tan^n x + 1}, \]

we can follow these steps:

### Step 1: Change the limits of integration
First, we rewrite the limits in terms of a common denominator. We can express \(\frac{\pi}{5}\) and \(\frac{3\pi}{10}\) with a denominator of 10:

\[ \frac{\pi}{5} = \frac{2\pi}{10}, \quad \frac{3\pi}{10} = \frac{3\pi}{10}. \]

Thus, we can rewrite the integral as:

\[ I = \int_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} \frac{dx}{\tan^n x + 1}. \]

### Step 2: Use the property of integrals
We will use the property of integrals that states:

\[ \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx. \]

In our case, \(a = \frac{2\pi}{10}\) and \(b = \frac{3\pi}{10}\). Therefore, we have:

\[ I = \int_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} \frac{dx}{\tan^n\left(\frac{5\pi}{10} - x\right) + 1}. \]

### Step 3: Simplify the integrand
Next, we simplify \(\tan\left(\frac{5\pi}{10} - x\right)\):

\[ \tan\left(\frac{5\pi}{10} - x\right) = \tan\left(\frac{\pi}{2} - x\right) = \cot x. \]

Thus, the integral becomes:

\[ I = \int_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} \frac{dx}{\cot^n x + 1}. \]

### Step 4: Rewrite cotangent in terms of tangent
We know that \(\cot x = \frac{1}{\tan x}\), so we can rewrite the integral as:

\[ I = \int_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} \frac{dx}{\left(\frac{1}{\tan x}\right)^n + 1} = \int_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} \frac{\tan^n x}{1 + \tan^n x} \, dx. \]

### Step 5: Combine the two integrals
Now we have two expressions for \(I\):

1. \(I = \int_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} \frac{dx}{\tan^n x + 1}\)
2. \(I = \int_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} \frac{\tan^n x}{1 + \tan^n x} \, dx\)

Adding these two equations gives:

\[ 2I = \int_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} \frac{1 + \tan^n x}{\tan^n x + 1} \, dx = \int_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} dx. \]

### Step 6: Evaluate the integral
The integral on the right simplifies to:

\[ \int_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} dx = \left[x\right]_{\frac{2\pi}{10}}^{\frac{3\pi}{10}} = \frac{3\pi}{10} - \frac{2\pi}{10} = \frac{\pi}{10}. \]

Thus, we have:

\[ 2I = \frac{\pi}{10} \implies I = \frac{\pi}{20}. \]

### Final Answer
Therefore, the value of the integral is:

\[ I = \frac{\pi}{20}. \]

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