`int sqrt(1+sin((x)/(4)))` dx is equal to

Options

A. `8(sin""(x)/(8)+cos""(x)/(8))+c`
B. `8(sin""(x)/(8)-cos""(x)/(8))+c`
C. `8(cos"'(x)/(8)-sin""(x)/(8))+c`
D. `(1)/(8)(sin""(x)/(8)-cos""(x)/(8))+c`

Explanation

To solve the integral \( \int \sqrt{1 + \sin\left(\frac{x}{4}\right)} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \sqrt{1 + \sin\left(\frac{x}{4}\right)} \, dx \]

### Step 2: Use a Trigonometric Identity
We know that:
\[ 1 + \sin\theta = \left(\sin\frac{\theta}{2} + \cos\frac{\theta}{2}\right)^2 \]
Thus, we can express:
\[ \sqrt{1 + \sin\left(\frac{x}{4}\right)} = \sqrt{\left(\sin\left(\frac{x}{8}\right) + \cos\left(\frac{x}{8}\right)\right)^2} = \sin\left(\frac{x}{8}\right) + \cos\left(\frac{x}{8}\right) \]

### Step 3: Substitute and Simplify
Now, substitute this back into the integral:
\[ \int \left(\sin\left(\frac{x}{8}\right) + \cos\left(\frac{x}{8}\right)\right) \, dx \]

### Step 4: Integrate Each Term
Now we can integrate each term separately:
1. For \( \sin\left(\frac{x}{8}\right) \):
\[ \int \sin\left(\frac{x}{8}\right) \, dx = -8 \cos\left(\frac{x}{8}\right) + C_1 \]
2. For \( \cos\left(\frac{x}{8}\right) \):
\[ \int \cos\left(\frac{x}{8}\right) \, dx = 8 \sin\left(\frac{x}{8}\right) + C_2 \]

### Step 5: Combine the Results
Combining the results from both integrals, we have:
\[ \int \sqrt{1 + \sin\left(\frac{x}{4}\right)} \, dx = -8 \cos\left(\frac{x}{8}\right) + 8 \sin\left(\frac{x}{8}\right) + C \]

### Final Result
Thus, the final answer is:
\[ \int \sqrt{1 + \sin\left(\frac{x}{4}\right)} \, dx = 8 \left(\sin\left(\frac{x}{8}\right) - \cos\left(\frac{x}{8}\right)\right) + C \]

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