What is `int_0^(4 pi) |cos x|dx` equal to

Options

A. 0
B. 2
C. 4
D. 8

Answer

To solve the integral \( \int_0^{4\pi} |\cos x| \, dx \), we can follow these steps:

### Step 1: Understand the function \( |\cos x| \)
The function \( |\cos x| \) is the absolute value of \( \cos x \). This means that wherever \( \cos x \) is negative, the graph of \( |\cos x| \) will reflect above the x-axis.

### Step 2: Determine the period of \( |\cos x| \)
The function \( \cos x \) has a period of \( 2\pi \). Therefore, \( |\cos x| \) also has a period of \( 2\pi \). This means that the integral from \( 0 \) to \( 4\pi \) can be broken down into two intervals of \( 0 \) to \( 2\pi \).

### Step 3: Break down the integral
We can express the integral as:
\[ \int_0^{4\pi} |\cos x| \, dx = \int_0^{2\pi} |\cos x| \, dx + \int_{2\pi}^{4\pi} |\cos x| \, dx \]
Since \( |\cos x| \) is periodic, both integrals will yield the same result:
\[ \int_0^{4\pi} |\cos x| \, dx = 2 \int_0^{2\pi} |\cos x| \, dx \]

### Step 4: Evaluate \( \int_0^{2\pi} |\cos x| \, dx \)
Now, we need to evaluate \( \int_0^{2\pi} |\cos x| \, dx \). The function \( \cos x \) is positive in the interval \( [0, \pi/2] \) and \( [3\pi/2, 2\pi] \), and negative in \( [\pi/2, 3\pi/2] \). Therefore, we can split the integral into three parts:
\[ \int_0^{2\pi} |\cos x| \, dx = \int_0^{\pi/2} \cos x \, dx + \int_{\pi/2}^{3\pi/2} -\cos x \, dx + \int_{3\pi/2}^{2\pi} \cos x \, dx \]

### Step 5: Calculate each integral
1. **First Integral**:
\[ \int_0^{\pi/2} \cos x \, dx = [\sin x]_0^{\pi/2} = \sin(\pi/2) - \sin(0) = 1 - 0 = 1 \]

2. **Second Integral**:
\[ \int_{\pi/2}^{3\pi/2} -\cos x \, dx = -[\sin x]_{\pi/2}^{3\pi/2} = -(\sin(3\pi/2) - \sin(\pi/2)) = -(-1 - 1) = 2 \]

3. **Third Integral**:
\[ \int_{3\pi/2}^{2\pi} \cos x \, dx = [\sin x]_{3\pi/2}^{2\pi} = \sin(2\pi) - \sin(3\pi/2) = 0 - (-1) = 1 \]

### Step 6: Combine the results
Now, we can combine the results of the three integrals:
\[ \int_0^{2\pi} |\cos x| \, dx = 1 + 2 + 1 = 4 \]

### Step 7: Final calculation
Now, substituting back into our earlier equation:
\[ \int_0^{4\pi} |\cos x| \, dx = 2 \int_0^{2\pi} |\cos x| \, dx = 2 \times 4 = 8 \]

### Final Answer
Thus, the value of \( \int_0^{4\pi} |\cos x| \, dx \) is \( 8 \).

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