Evaluate : `int_(0)^((pi)/(4)) sec^(2) x dx `

Answer

To evaluate the integral \( \int_{0}^{\frac{\pi}{4}} \sec^2 x \, dx \), we can follow these steps:

### Step 1: Identify the integral
We need to evaluate the integral:
\[ \int_{0}^{\frac{\pi}{4}} \sec^2 x \, dx \]

### Step 2: Recall the antiderivative
The antiderivative of \( \sec^2 x \) is:
\[ \int \sec^2 x \, dx = \tan x + C \]
where \( C \) is the constant of integration.

### Step 3: Apply the limits of integration
Now we will apply the limits from \( 0 \) to \( \frac{\pi}{4} \):
\[ \left[ \tan x \right]_{0}^{\frac{\pi}{4}} = \tan\left(\frac{\pi}{4}\right) - \tan(0) \]

### Step 4: Calculate the values of the tangent function
We know that:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \quad \text{and} \quad \tan(0) = 0 \]
Thus, substituting these values gives:
\[ \tan\left(\frac{\pi}{4}\right) - \tan(0) = 1 - 0 = 1 \]

### Step 5: Final answer
Therefore, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{4}} \sec^2 x \, dx = 1 \]