The value of the definite integral `int _(3)^(7) (cos x ^(2))/(cos x ^(2) + cos ((10) -x )^(2)dx` is:

Options

A. 2
B. 1
C. `1/2`
D. None of these

Answer

To solve the definite integral

\[ I = \int_{3}^{7} \frac{\cos^2 x}{\cos^2 x + \cos^2(10 - x)} \, dx, \]

we can use a property of definite integrals. This 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 apply this property to our integral. Here, \( a = 3 \) and \( b = 7 \), so \( a + b = 10 \). Thus, we can rewrite the integral as:

\[ I = \int_{3}^{7} \frac{\cos^2(10 - x)}{\cos^2(10 - x) + \cos^2 x} \, dx. \]

### Step 2: Rewrite the integral

Now, we can express this new integral as:

\[ I = \int_{3}^{7} \frac{\cos^2(10 - x)}{\cos^2(10 - x) + \cos^2 x} \, dx. \]

### Step 3: Add the two integrals

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

1. \( I = \int_{3}^{7} \frac{\cos^2 x}{\cos^2 x + \cos^2(10 - x)} \, dx \) (Equation 1)
2. \( I = \int_{3}^{7} \frac{\cos^2(10 - x)}{\cos^2(10 - x) + \cos^2 x} \, dx \) (Equation 2)

Adding both equations gives:

\[ 2I = \int_{3}^{7} \left( \frac{\cos^2 x}{\cos^2 x + \cos^2(10 - x)} + \frac{\cos^2(10 - x)}{\cos^2(10 - x) + \cos^2 x} \right) dx. \]

### Step 4: Simplify the integrand

The integrand simplifies as follows:

\[ \frac{\cos^2 x + \cos^2(10 - x)}{\cos^2 x + \cos^2(10 - x)} = 1. \]

Thus, we have:

\[ 2I = \int_{3}^{7} 1 \, dx. \]

### Step 5: Evaluate the integral

Now we can evaluate the integral:

\[ \int_{3}^{7} 1 \, dx = x \bigg|_{3}^{7} = 7 - 3 = 4. \]

### Step 6: Solve for \( I \)

Now we can solve for \( I \):

\[ 2I = 4 \implies I = \frac{4}{2} = 2. \]

Thus, the value of the definite integral is

\[ \boxed{2}. \]