Evaluate : `int_(0)^((pi)/(4)) log(1+tan theta ) d theta`

Answer

To evaluate the integral

\[ I = \int_0^{\frac{\pi}{4}} \log(1 + \tan \theta) \, d\theta, \]

we will use a property of definite integrals.

### Step 1: Apply the property of definite integrals

We can use the property that states:

\[ \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx. \]

In our case, we set \( a = \frac{\pi}{4} \) and \( f(\theta) = \log(1 + \tan \theta) \). Thus, we have:

\[ I = \int_0^{\frac{\pi}{4}} \log(1 + \tan(\frac{\pi}{4} - \theta)) \, d\theta. \]

### Step 2: Simplify \( \tan(\frac{\pi}{4} - \theta) \)

Using the tangent subtraction formula, we find:

\[ \tan\left(\frac{\pi}{4} - \theta\right) = \frac{\tan\left(\frac{\pi}{4}\right) - \tan(\theta)}{1 + \tan\left(\frac{\pi}{4}\right)\tan(\theta)} = \frac{1 - \tan(\theta)}{1 + \tan(\theta)}. \]

### Step 3: Substitute back into the integral

Now, we substitute this back into the integral:

\[ I = \int_0^{\frac{\pi}{4}} \log\left(1 + \frac{1 - \tan \theta}{1 + \tan \theta}\right) \, d\theta. \]

### Step 4: Simplify the logarithm

We can simplify the expression inside the logarithm:

\[ 1 + \frac{1 - \tan \theta}{1 + \tan \theta} = \frac{(1 + \tan \theta) + (1 - \tan \theta)}{1 + \tan \theta} = \frac{2}{1 + \tan \theta}. \]

Thus, we have:

\[ I = \int_0^{\frac{\pi}{4}} \log\left(\frac{2}{1 + \tan \theta}\right) \, d\theta. \]

### Step 5: Split the logarithm

Using the property of logarithms, we can split this into two integrals:

\[ I = \int_0^{\frac{\pi}{4}} \log(2) \, d\theta - \int_0^{\frac{\pi}{4}} \log(1 + \tan \theta) \, d\theta. \]

### Step 6: Evaluate the first integral

The first integral is straightforward:

\[ \int_0^{\frac{\pi}{4}} \log(2) \, d\theta = \log(2) \cdot \left(\frac{\pi}{4} - 0\right) = \frac{\pi}{4} \log(2). \]

### Step 7: Combine the results

Now we have:

\[ I = \frac{\pi}{4} \log(2) - I. \]

### Step 8: Solve for \( I \)

Adding \( I \) to both sides gives:

\[ 2I = \frac{\pi}{4} \log(2). \]

Dividing by 2, we find:

\[ I = \frac{\pi}{8} \log(2). \]

### Final Result

Thus, the value of the integral is:

\[ \int_0^{\frac{\pi}{4}} \log(1 + \tan \theta) \, d\theta = \frac{\pi}{8} \log(2). \]

---