The value of `int_(0)^(2) |"cos"(pi)/(2)x|dx` is

Options

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

Answer

To solve the integral \( \int_{0}^{2} |\cos\left(\frac{\pi}{2} x\right)| \, dx \), we will follow these steps:

### Step 1: Identify the function and its behavior
The function we are integrating is \( |\cos\left(\frac{\pi}{2} x\right)| \). We need to determine where this function changes sign within the interval \([0, 2]\).

### Step 2: Find the points where the function is zero
To find where \( \cos\left(\frac{\pi}{2} x\right) = 0 \):
\[ \frac{\pi}{2} x = \frac{\pi}{2} + n\pi \quad \text{for } n \in \mathbb{Z} \]
This gives:
\[ x = 1 + 2n \quad \text{for } n \in \mathbb{Z} \]
Within the interval \([0, 2]\), the relevant point is \( x = 1 \).

### Step 3: Break the integral at the point where the function changes sign
Since \( \cos\left(\frac{\pi}{2} x\right) \) is positive on \([0, 1]\) and negative on \([1, 2]\), we can express the integral as:
\[ \int_{0}^{2} |\cos\left(\frac{\pi}{2} x\right)| \, dx = \int_{0}^{1} \cos\left(\frac{\pi}{2} x\right) \, dx + \int_{1}^{2} -\cos\left(\frac{\pi}{2} x\right) \, dx \]

### Step 4: Calculate the first integral
For the first integral:
\[ \int_{0}^{1} \cos\left(\frac{\pi}{2} x\right) \, dx \]
Using the integral formula \( \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \):
\[ = \left[ \frac{2}{\pi} \sin\left(\frac{\pi}{2} x\right) \right]_{0}^{1} = \frac{2}{\pi} \left( \sin\left(\frac{\pi}{2}\right) - \sin(0) \right) = \frac{2}{\pi} (1 - 0) = \frac{2}{\pi} \]

### Step 5: Calculate the second integral
For the second integral:
\[ \int_{1}^{2} -\cos\left(\frac{\pi}{2} x\right) \, dx \]
This becomes:
\[ -\left[ \frac{2}{\pi} \sin\left(\frac{\pi}{2} x\right) \right]_{1}^{2} = -\frac{2}{\pi} \left( \sin(\pi) - \sin\left(\frac{\pi}{2}\right) \right) = -\frac{2}{\pi} (0 - 1) = \frac{2}{\pi} \]

### Step 6: Combine the results
Now, we combine both integrals:
\[ \int_{0}^{2} |\cos\left(\frac{\pi}{2} x\right)| \, dx = \frac{2}{\pi} + \frac{2}{\pi} = \frac{4}{\pi} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{0}^{2} |\cos\left(\frac{\pi}{2} x\right)| \, dx = \frac{4}{\pi} \]

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