`int_(0)^(50pi)| cos x|dx=`

Options

A. 100
B. 50
C. 0
D. none of these

Answer

To solve the integral \( I = \int_{0}^{50\pi} |\cos x| \, dx \), we can follow these steps:

### Step 1: Use the periodic property of the cosine function
The function \( |\cos x| \) is periodic with a period of \( 2\pi \). Therefore, we can express the integral over the interval \( [0, 50\pi] \) in terms of the integral over one period \( [0, 2\pi] \).

\[ I = \int_{0}^{50\pi} |\cos x| \, dx = 25 \int_{0}^{2\pi} |\cos x| \, dx \]

### Step 2: Break down the integral over one period
Next, we need to evaluate \( \int_{0}^{2\pi} |\cos x| \, dx \). We can break this integral into two parts where \( \cos x \) is positive and negative:

- From \( 0 \) to \( \pi/2 \), \( \cos x \) is positive.
- From \( \pi/2 \) to \( \pi \), \( \cos x \) is negative.
- From \( \pi \) to \( 3\pi/2 \), \( \cos x \) is negative.
- From \( 3\pi/2 \) to \( 2\pi \), \( \cos x \) is positive.

Thus, we can write:

\[ \int_{0}^{2\pi} |\cos x| \, dx = \int_{0}^{\pi/2} \cos x \, dx + \int_{\pi/2}^{\pi} -\cos x \, dx + \int_{\pi}^{3\pi/2} -\cos x \, dx + \int_{3\pi/2}^{2\pi} \cos x \, dx \]

### Step 3: Evaluate each integral
Now we evaluate each integral:

1. \( \int_{0}^{\pi/2} \cos x \, dx = \left[ \sin x \right]_{0}^{\pi/2} = \sin(\pi/2) - \sin(0) = 1 - 0 = 1 \)

2. \( \int_{\pi/2}^{\pi} -\cos x \, dx = -\left[ \sin x \right]_{\pi/2}^{\pi} = -(\sin(\pi) - \sin(\pi/2)) = -(0 - 1) = 1 \)

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

4. \( \int_{3\pi/2}^{2\pi} \cos x \, dx = \left[ \sin x \right]_{3\pi/2}^{2\pi} = \sin(2\pi) - \sin(3\pi/2) = 0 - (-1) = 1 \)

### Step 4: Combine the results
Now we can combine these results:

\[ \int_{0}^{2\pi} |\cos x| \, dx = 1 + 1 + 1 + 1 = 4 \]

### Step 5: Calculate the final result
Substituting back into the expression for \( I \):

\[ I = 25 \int_{0}^{2\pi} |\cos x| \, dx = 25 \times 4 = 100 \]

### Final Answer
Thus, the value of the integral is:

\[ \int_{0}^{50\pi} |\cos x| \, dx = 100 \]