`int_(0)^(2pi)e^(x//2)sin((x)/(2)+(pi)/(4))dx=`

Options

A. `2pi`
B. `e^(pi)`
C. 0
D. `2sqrt(2)`

Explanation

To solve the integral

\[ \int_{0}^{2\pi} e^{\frac{x}{2}} \sin\left(\frac{x}{2} + \frac{\pi}{4}\right) dx, \]

we will follow these steps:

### Step 1: Substitution
Let \( t = \frac{x}{2} \). Then, \( x = 2t \) and \( dx = 2 dt \). The limits change as follows:
- When \( x = 0 \), \( t = 0 \).
- When \( x = 2\pi \), \( t = \pi \).

Thus, the integral becomes:

\[ \int_{0}^{\pi} e^{t} \sin\left(t + \frac{\pi}{4}\right) \cdot 2 dt = 2 \int_{0}^{\pi} e^{t} \sin\left(t + \frac{\pi}{4}\right) dt. \]

### Step 2: Expand the Sine Function
Using the angle addition formula for sine, we have:

\[ \sin\left(t + \frac{\pi}{4}\right) = \sin t \cos\left(\frac{\pi}{4}\right) + \cos t \sin\left(\frac{\pi}{4}\right). \]

Since \( \cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \), we can rewrite the sine function as:

\[ \sin\left(t + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \sin t + \frac{1}{\sqrt{2}} \cos t. \]

### Step 3: Substitute Back into the Integral
Substituting this back into the integral gives:

\[ 2 \int_{0}^{\pi} e^{t} \left(\frac{1}{\sqrt{2}} \sin t + \frac{1}{\sqrt{2}} \cos t\right) dt = \frac{2}{\sqrt{2}} \int_{0}^{\pi} e^{t} \sin t dt + \frac{2}{\sqrt{2}} \int_{0}^{\pi} e^{t} \cos t dt. \]

### Step 4: Factor Out Constants
Factoring out the constant \( \frac{2}{\sqrt{2}} \):

\[ \frac{2}{\sqrt{2}} \left( \int_{0}^{\pi} e^{t} \sin t dt + \int_{0}^{\pi} e^{t} \cos t dt \right). \]

### Step 5: Evaluate the Integrals
To evaluate the integrals \( \int e^{t} \sin t dt \) and \( \int e^{t} \cos t dt \), we can use integration by parts or the known result:

\[ \int e^{t} \sin t dt = e^{t} \left(\sin t - \cos t\right) + C, \]
\[ \int e^{t} \cos t dt = e^{t} \left(\cos t + \sin t\right) + C. \]

### Step 6: Apply Limits
Evaluating these from \( 0 \) to \( \pi \):

1. For \( \int_{0}^{\pi} e^{t} \sin t dt \):
- At \( t = \pi \): \( e^{\pi}(\sin \pi - \cos \pi) = e^{\pi}(0 + 1) = e^{\pi} \).
- At \( t = 0 \): \( e^{0}(\sin 0 - \cos 0) = 1(0 - 1) = -1 \).
- Thus, \( \int_{0}^{\pi} e^{t} \sin t dt = e^{\pi} + 1 \).

2. For \( \int_{0}^{\pi} e^{t} \cos t dt \):
- At \( t = \pi \): \( e^{\pi}(\cos \pi + \sin \pi) = e^{\pi}(-1 + 0) = -e^{\pi} \).
- At \( t = 0 \): \( e^{0}(\cos 0 + \sin 0) = 1(1 + 0) = 1 \).
- Thus, \( \int_{0}^{\pi} e^{t} \cos t dt = -e^{\pi} + 1 \).

### Step 7: Combine Results
Combining these results:

\[ \int_{0}^{\pi} e^{t} \sin t dt + \int_{0}^{\pi} e^{t} \cos t dt = (e^{\pi} + 1) + (-e^{\pi} + 1) = 2. \]

### Final Calculation
Now substituting back into our expression:

\[ \frac{2}{\sqrt{2}} \cdot 2 = \frac{4}{\sqrt{2}} = 2\sqrt{2}. \]

Thus, the final answer is:

\[ \int_{0}^{2\pi} e^{\frac{x}{2}} \sin\left(\frac{x}{2} + \frac{\pi}{4}\right) dx = 2\sqrt{2}. \]