What is `int_0^(pi//2) (d theta)/(1+ cos theta)` equal to

Options

A. `1/2`
B. 1
C. `sqrt3`
D. none of these

Answer

To solve the integral \( \int_0^{\frac{\pi}{2}} \frac{d\theta}{1 + \cos \theta} \), we can follow these steps:

### Step 1: Rewrite the Denominator
We start by rewriting the expression \( 1 + \cos \theta \). Using the double angle identity, we have:
\[ 1 + \cos \theta = 1 + 2 \cos^2\left(\frac{\theta}{2}\right) - 1 = 2 \cos^2\left(\frac{\theta}{2}\right) \]
Thus, we can rewrite our integral as:
\[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{1 + \cos \theta} = \int_0^{\frac{\pi}{2}} \frac{d\theta}{2 \cos^2\left(\frac{\theta}{2}\right)} \]

### Step 2: Factor Out the Constant
Next, we can factor out the constant \( \frac{1}{2} \) from the integral:
\[ = \frac{1}{2} \int_0^{\frac{\pi}{2}} \sec^2\left(\frac{\theta}{2}\right) d\theta \]

### Step 3: Change of Variables
Now, we will perform a substitution to simplify the integral. Let:
\[ u = \frac{\theta}{2} \quad \Rightarrow \quad d\theta = 2 du \]
When \( \theta = 0 \), \( u = 0 \) and when \( \theta = \frac{\pi}{2} \), \( u = \frac{\pi}{4} \). Thus, the integral becomes:
\[ = \frac{1}{2} \int_0^{\frac{\pi}{4}} \sec^2(u) \cdot 2 du = \int_0^{\frac{\pi}{4}} \sec^2(u) du \]

### Step 4: Integrate
The integral of \( \sec^2(u) \) is:
\[ \int \sec^2(u) du = \tan(u) \]
So we evaluate:
\[ = \left[ \tan(u) \right]_0^{\frac{\pi}{4}} = \tan\left(\frac{\pi}{4}\right) - \tan(0) = 1 - 0 = 1 \]

### Final Result
Thus, the value of the integral \( \int_0^{\frac{\pi}{2}} \frac{d\theta}{1 + \cos \theta} \) is:
\[ \boxed{1} \]