The value of `int_(0)^(32pi//3) sqrt(1+cos2x)dx` is

Options

A. `(44+sqrt(3))/(sqrt(2))`
B. `(44-sqrt(3))/(sqrt(2))`
C. `(22-sqrt(3))/(sqrt(2))`
D. `(22+sqrt(3))/(sqrt(2))`

Explanation

To solve the integral \( I = \int_{0}^{\frac{32\pi}{3}} \sqrt{1 + \cos 2x} \, dx \), we can follow these steps:

### Step 1: Simplify the integrand
We know that:
\[ 1 + \cos 2x = 2 \cos^2 x \]
Thus, we can rewrite the integrand:
\[ \sqrt{1 + \cos 2x} = \sqrt{2 \cos^2 x} = \sqrt{2} |\cos x| \]
So, the integral becomes:
\[ I = \int_{0}^{\frac{32\pi}{3}} \sqrt{2} |\cos x| \, dx \]

### Step 2: Factor out the constant
We can factor out \(\sqrt{2}\):
\[ I = \sqrt{2} \int_{0}^{\frac{32\pi}{3}} |\cos x| \, dx \]

### Step 3: Determine the period of \(|\cos x|\)
The function \(|\cos x|\) has a period of \(\pi\). We can find how many complete periods fit into the interval \([0, \frac{32\pi}{3}]\):
\[ \frac{32\pi/3}{\pi} = \frac{32}{3} = 10 + \frac{2}{3} \]
This means there are 10 complete periods of \(|\cos x|\) and an additional \(\frac{2}{3}\) of a period.

### Step 4: Break the integral into parts
We can express the integral as:
\[ I = \sqrt{2} \left( \int_{0}^{10\pi} |\cos x| \, dx + \int_{10\pi}^{\frac{32\pi}{3}} |\cos x| \, dx \right) \]

### Step 5: Calculate the integral over one period
The integral of \(|\cos x|\) over one period \([0, \pi]\) is:
\[ \int_{0}^{\pi} |\cos x| \, dx = \int_{0}^{\pi} \cos x \, dx = [\sin x]_{0}^{\pi} = \sin(\pi) - \sin(0) = 0 - 0 = 2 \]
Thus, for 10 periods:
\[ \int_{0}^{10\pi} |\cos x| \, dx = 10 \cdot 2 = 20 \]

### Step 6: Calculate the remaining integral
Now we need to calculate:
\[ \int_{10\pi}^{\frac{32\pi}{3}} |\cos x| \, dx \]
The interval \([10\pi, \frac{32\pi}{3}]\) can be rewritten as:
\[ 10\pi \text{ to } 10\pi + \frac{2\pi}{3} \]
This corresponds to the interval \([0, \frac{2\pi}{3}]\) for \(|\cos x|\).

### Step 7: Calculate the integral from \(0\) to \(\frac{2\pi}{3}\)
\[ \int_{0}^{\frac{2\pi}{3}} |\cos x| \, dx = \int_{0}^{\frac{2\pi}{3}} \cos x \, dx = [\sin x]_{0}^{\frac{2\pi}{3}} = \sin\left(\frac{2\pi}{3}\right) - \sin(0) = \frac{\sqrt{3}}{2} - 0 = \frac{\sqrt{3}}{2} \]

### Step 8: Combine the results
Now we can combine everything:
\[ I = \sqrt{2} \left( 20 + \frac{\sqrt{3}}{2} \right) \]
\[ I = 20\sqrt{2} + \frac{\sqrt{6}}{2} \]

### Final Result
Thus, the value of the integral is:
\[ I = 20\sqrt{2} + \frac{\sqrt{6}}{2} \]