`int_(pi//4)^(3pi//4)(dx)/(1+cos x)=`

Options

A. 2
B. `-2`
C. `(1)/(2)`
D. `-(1)/(2)`

Answer

To solve the integral

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

we will use a property of definite integrals. The property states that

\[ \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx \]

### Step 1: Apply the property of definite integrals

Let’s denote the limits \( a = \frac{\pi}{4} \) and \( b = \frac{3\pi}{4} \). Thus, we have:

\[ a + b = \frac{\pi}{4} + \frac{3\pi}{4} = \pi \]

Using the property, we can rewrite the integral:

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

### Step 2: Simplify \( \cos(\pi - x) \)

Using the identity \( \cos(\pi - x) = -\cos x \), we can substitute this into our integral:

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

### Step 3: Add the two expressions for \( I \)

Now we have two expressions for \( I \):

1. \( I = \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x} \)
2. \( I = \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 - \cos x} \)

Adding these two equations:

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

### Step 4: Combine the fractions

To combine the fractions, we find a common denominator:

\[ \frac{1}{1 + \cos x} + \frac{1}{1 - \cos x} = \frac{(1 - \cos x) + (1 + \cos x)}{(1 + \cos x)(1 - \cos x)} = \frac{2}{1 - \cos^2 x} \]

Using the identity \( 1 - \cos^2 x = \sin^2 x \):

\[ \frac{2}{1 - \cos^2 x} = \frac{2}{\sin^2 x} \]

### Step 5: Substitute back into the integral

Thus, we can rewrite our equation for \( 2I \):

\[ 2I = \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{2}{\sin^2 x} \, dx \]

This simplifies to:

\[ 2I = 2 \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \csc^2 x \, dx \]

### Step 6: Integrate \( \csc^2 x \)

The integral of \( \csc^2 x \) is \( -\cot x \):

\[ 2I = 2 \left[ -\cot x \right]_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \]

### Step 7: Evaluate the limits

Now we evaluate:

\[ -\cot\left(\frac{3\pi}{4}\right) + \cot\left(\frac{\pi}{4}\right) \]

We know:

- \( \cot\left(\frac{3\pi}{4}\right) = -1 \)
- \( \cot\left(\frac{\pi}{4}\right) = 1 \)

Thus, we have:

\[ 2I = 2 \left[ -(-1) + 1 \right] = 2 \left[ 1 + 1 \right] = 4 \]

### Step 8: Solve for \( I \)

Dividing by 2:

\[ I = 2 \]

### Final Answer

Thus, the value of the integral is:

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