The value of `I=int_(0)^(pi//2) (1)/(1+cosx)dx` is

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. 1
D. `pi`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \cos x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Denominator
We can use the trigonometric identity \( \cos x = 2 \cos^2\left(\frac{x}{2}\right) - 1 \) to rewrite the denominator:
\[ 1 + \cos x = 1 + (2 \cos^2\left(\frac{x}{2}\right) - 1) = 2 \cos^2\left(\frac{x}{2}\right) \]
Thus, we can rewrite the integral as:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{2 \cos^2\left(\frac{x}{2}\right)} \, dx \]

### Step 2: Factor Out the Constant
We can factor out the constant \( \frac{1}{2} \) from the integral:
\[ I = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \sec^2\left(\frac{x}{2}\right) \, dx \]

### Step 3: Change of Variable
Next, we perform a change of variable. Let \( u = \frac{x}{2} \), then \( dx = 2 \, du \). The limits change as follows:
- When \( x = 0 \), \( u = 0 \)
- When \( x = \frac{\pi}{2} \), \( u = \frac{\pi}{4} \)

Now, we can rewrite the integral:
\[ I = \frac{1}{2} \int_{0}^{\frac{\pi}{4}} \sec^2(u) \cdot 2 \, du = \int_{0}^{\frac{\pi}{4}} \sec^2(u) \, du \]

### Step 4: Integrate
The integral of \( \sec^2(u) \) is \( \tan(u) \):
\[ I = \left[ \tan(u) \right]_{0}^{\frac{\pi}{4}} = \tan\left(\frac{\pi}{4}\right) - \tan(0) \]

### Step 5: Evaluate the Limits
Now we evaluate the limits:
\[ I = \tan\left(\frac{\pi}{4}\right) - \tan(0) = 1 - 0 = 1 \]

### Final Answer
Thus, the value of the integral \( I \) is:
\[ \boxed{1} \]