What is `int_(0)^(pi/4) (d theta)/(1+cos theta)` equal to ?

Options

A. `1/2`
B. `1`
C. `sqrt(3)`
D. None of the above

Explanation

To solve the integral \( \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{1 + \cos \theta} \), we will follow these steps:

### Step 1: Rewrite the integral
We start with the integral:
\[ \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{1 + \cos \theta} \]

### Step 2: Use the identity for \(1 + \cos \theta\)
We can use the trigonometric identity:
\[ 1 + \cos \theta = 2 \cos^2\left(\frac{\theta}{2}\right) \]
Thus, we can rewrite the integral as:
\[ \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{2 \cos^2\left(\frac{\theta}{2}\right)} \]

### Step 3: Factor out the constant
We can factor out the constant \( \frac{1}{2} \):
\[ \frac{1}{2} \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{\cos^2\left(\frac{\theta}{2}\right)} \]

### Step 4: Change of variable
Let \( u = \frac{\theta}{2} \), then \( d\theta = 2 du \). When \( \theta = 0 \), \( u = 0 \) and when \( \theta = \frac{\pi}{4} \), \( u = \frac{\pi}{8} \). The integral now becomes:
\[ \frac{1}{2} \int_{0}^{\frac{\pi}{8}} \frac{2 du}{\cos^2(u)} = \int_{0}^{\frac{\pi}{8}} \sec^2(u) du \]

### Step 5: Integrate \( \sec^2(u) \)
The integral of \( \sec^2(u) \) is:
\[ \int \sec^2(u) du = \tan(u) \]
Thus, we have:
\[ \int_{0}^{\frac{\pi}{8}} \sec^2(u) du = \tan\left(\frac{\pi}{8}\right) - \tan(0) \]
Since \( \tan(0) = 0 \), we get:
\[ \tan\left(\frac{\pi}{8}\right) \]

### Step 6: Final result
Thus, the value of the original integral is:
\[ \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{1 + \cos \theta} = \tan\left(\frac{\pi}{8}\right) \]

### Summary
The final answer is:
\[ \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{1 + \cos \theta} = \tan\left(\frac{\pi}{8}\right) \]