If `f(x) =|{:(cosx,1,0),(1,2 cosx,1),(0,1,2 cosx):}|, "then" |overset(pi//2)underset(0)intf(x)dx|` is equal to

Options

A. `1//4`
B. `1//3`
C. `1//2`
D. 1

Answer

To solve the problem, we need to evaluate the integral of the determinant function \( f(x) \) defined as:

\[ f(x) = \left| \begin{array}{ccc} \cos x & 1 & 0 \\ 1 & 2 \cos x & 1 \\ 0 & 1 & 2 \cos x \end{array} \right| \]

### Step 1: Calculate the Determinant

We will calculate the determinant using the formula for a \( 3 \times 3 \) matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:

\[ A = \begin{bmatrix} \cos x & 1 & 0 \\ 1 & 2 \cos x & 1 \\ 0 & 1 & 2 \cos x \end{bmatrix} \]

Here, \( a = \cos x, b = 1, c = 0, d = 1, e = 2 \cos x, f = 1, g = 0, h = 1, i = 2 \cos x \).

Now, we can compute the determinant:

\[ \text{det}(A) = \cos x \left( (2 \cos x)(2 \cos x) - (1)(1) \right) - 1 \left( (1)(2 \cos x) - (1)(0) \right) + 0 \]

Calculating the terms:

1. \( (2 \cos x)(2 \cos x) - 1 = 4 \cos^2 x - 1 \)
2. \( (1)(2 \cos x) - (1)(0) = 2 \cos x \)

Putting it all together:

\[ \text{det}(A) = \cos x (4 \cos^2 x - 1) - 2 \cos x \]

This simplifies to:

\[ \text{det}(A) = 4 \cos^3 x - \cos x - 2 \cos x = 4 \cos^3 x - 3 \cos x \]

Thus,

\[ f(x) = 4 \cos^3 x - 3 \cos x \]

### Step 2: Set Up the Integral

Now we need to evaluate the integral:

\[ \int_0^{\frac{\pi}{2}} f(x) \, dx = \int_0^{\frac{\pi}{2}} (4 \cos^3 x - 3 \cos x) \, dx \]

### Step 3: Split the Integral

We can split the integral into two parts:

\[ \int_0^{\frac{\pi}{2}} f(x) \, dx = 4 \int_0^{\frac{\pi}{2}} \cos^3 x \, dx - 3 \int_0^{\frac{\pi}{2}} \cos x \, dx \]

### Step 4: Evaluate Each Integral

1. **Integral of \( \cos x \)**:

\[ \int_0^{\frac{\pi}{2}} \cos x \, dx = [\sin x]_0^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1 \]

2. **Integral of \( \cos^3 x \)**:

To evaluate \( \int_0^{\frac{\pi}{2}} \cos^3 x \, dx \), we can use the reduction formula or a known result:

\[ \int_0^{\frac{\pi}{2}} \cos^3 x \, dx = \frac{2}{3} \]

### Step 5: Substitute Back

Now substituting back into the integral:

\[ \int_0^{\frac{\pi}{2}} f(x) \, dx = 4 \left(\frac{2}{3}\right) - 3(1) = \frac{8}{3} - 3 = \frac{8}{3} - \frac{9}{3} = -\frac{1}{3} \]

### Step 6: Take the Absolute Value

Finally, we take the absolute value:

\[ \left| \int_0^{\frac{\pi}{2}} f(x) \, dx \right| = \left| -\frac{1}{3} \right| = \frac{1}{3} \]

### Final Answer

Thus, the value of the integral is:

\[ \frac{1}{3} \]