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 compute the determinant \( f(x) \):

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

Calculating the 2x2 determinants:

1. For the first determinant:
\[ \left| \begin{array}{cc} 2 \cos x & 1 \\ 1 & 2 \cos x \end{array} \right| = (2 \cos x)(2 \cos x) - (1)(1) = 4 \cos^2 x - 1 \]

2. For the second determinant:
\[ \left| \begin{array}{cc} 1 & 1 \\ 0 & 2 \cos x \end{array} \right| = (1)(2 \cos x) - (1)(0) = 2 \cos x \]

Now substituting back into \( f(x) \):

\[ f(x) = \cos x (4 \cos^2 x - 1) - 2 \cos x \]
\[ = 4 \cos^3 x - \cos x - 2 \cos x \]
\[ = 4 \cos^3 x - 3 \cos x \]

### Step 2: Recognize the Trigonometric Identity

Notice that \( 4 \cos^3 x - 3 \cos x \) can be rewritten using the cosine triple angle identity:

\[ f(x) = \cos(3x) \]

### Step 3: Set Up the Integral

We need to evaluate the integral:

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

### Step 4: Evaluate the Integral

The integral of \( \cos(3x) \) is:

\[ \int \cos(3x) \, dx = \frac{1}{3} \sin(3x) + C \]

Now, we evaluate it from 0 to \( \frac{\pi}{2} \):

\[ \int_0^{\frac{\pi}{2}} \cos(3x) \, dx = \left[ \frac{1}{3} \sin(3x) \right]_0^{\frac{\pi}{2}} = \frac{1}{3} \sin\left(3 \cdot \frac{\pi}{2}\right) - \frac{1}{3} \sin(0) \]

Calculating the sine values:

\[ = \frac{1}{3} \sin\left(\frac{3\pi}{2}\right) - 0 = \frac{1}{3} \cdot (-1) = -\frac{1}{3} \]

### Step 5: Take the Absolute Value

Since the problem asks for 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 final answer is:

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