`int_(pi //4)^(3pi//4) (dx)/( 1+ cosx) ` is equal to

Options

A. `-2`
B. 2
C. 4
D. `-4`

Answer

To solve the integral

\[ I = \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}, \]

we will follow these steps:

### Step 1: Simplify the integrand
We can use the identity \(1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right)\). Thus, we can rewrite the integrand:

\[ \frac{1}{1 + \cos x} = \frac{1}{2 \cos^2\left(\frac{x}{2}\right)} = \frac{1}{2} \sec^2\left(\frac{x}{2}\right). \]

### Step 2: Rewrite the integral
Substituting this back into the integral, we have:

\[ I = \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{1}{2} \sec^2\left(\frac{x}{2}\right) dx. \]

### Step 3: Change of variables
Let \(u = \frac{x}{2}\), then \(dx = 2 du\). The limits change as follows:
- When \(x = \frac{\pi}{4}\), \(u = \frac{\pi}{8}\).
- When \(x = \frac{3\pi}{4}\), \(u = \frac{3\pi}{8}\).

Thus, the integral becomes:

\[ I = \int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} \sec^2(u) du. \]

### Step 4: Integrate
The integral of \(\sec^2(u)\) is \(\tan(u)\):

\[ I = \left[ \tan(u) \right]_{\frac{\pi}{8}}^{\frac{3\pi}{8}}. \]

### Step 5: Evaluate the limits
Now we evaluate the limits:

\[ I = \tan\left(\frac{3\pi}{8}\right) - \tan\left(\frac{\pi}{8}\right). \]

### Step 6: Use the tangent addition formula
Using the identity \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\) with \(a = \frac{\pi}{4}\) and \(b = \frac{\pi}{8}\):

\[ \tan\left(\frac{3\pi}{8}\right) = \tan\left(\frac{\pi}{4} + \frac{\pi}{8}\right) = \frac{\tan\left(\frac{\pi}{4}\right) + \tan\left(\frac{\pi}{8}\right)}{1 - \tan\left(\frac{\pi}{4}\right) \tan\left(\frac{\pi}{8}\right)} = \frac{1 + \tan\left(\frac{\pi}{8}\right)}{1 - \tan\left(\frac{\pi}{8}\right)}. \]

### Step 7: Substitute back
Thus, we have:

\[ I = \frac{1 + \tan\left(\frac{\pi}{8}\right)}{1 - \tan\left(\frac{\pi}{8}\right)} - \tan\left(\frac{\pi}{8}\right). \]

### Step 8: Simplify
After simplification, we find:

\[ I = 2. \]

### Final Answer
Thus, the value of the integral is:

\[ \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x} = 2. \]