`int_(-pi//4)^(pi//4) e^(-x)sin x" dx"` is

Options

A. `-(sqrt(2))/(2)e^(-pi//4)`
B. `(sqrt(2))/(2)e^(-pi//4)`
C. `sqrt(2)(e^(-pi//4)-e^(pi//4))`
D. zero

Answer

To solve the integral \( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} e^{-x} \sin x \, dx \), we will use integration by parts. Here’s a step-by-step solution:

### Step 1: Set Up the Integral
Let
\[ I = \int e^{-x} \sin x \, dx \]

### Step 2: Apply Integration by Parts
Using integration by parts, we choose:
- \( u = \sin x \) and \( dv = e^{-x} dx \)
- Then, \( du = \cos x \, dx \) and \( v = -e^{-x} \)

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we have:
\[ I = -e^{-x} \sin x - \int -e^{-x} \cos x \, dx \]
This simplifies to:
\[ I = -e^{-x} \sin x + \int e^{-x} \cos x \, dx \]

### Step 3: Apply Integration by Parts Again
Now, we need to evaluate the integral \( \int e^{-x} \cos x \, dx \). We apply integration by parts again:
- Let \( u = \cos x \) and \( dv = e^{-x} dx \)
- Then, \( du = -\sin x \, dx \) and \( v = -e^{-x} \)

Applying integration by parts:
\[ \int e^{-x} \cos x \, dx = -e^{-x} \cos x - \int -e^{-x} (-\sin x) \, dx \]
This simplifies to:
\[ \int e^{-x} \cos x \, dx = -e^{-x} \cos x - \int e^{-x} \sin x \, dx \]
Thus, we can express this as:
\[ \int e^{-x} \cos x \, dx = -e^{-x} \cos x - I \]

### Step 4: Substitute Back
Substituting this back into our expression for \( I \):
\[ I = -e^{-x} \sin x - e^{-x} \cos x - I \]
Now, rearranging gives:
\[ 2I = -e^{-x} (\sin x + \cos x) \]
Thus,
\[ I = -\frac{1}{2} e^{-x} (\sin x + \cos x) \]

### Step 5: Evaluate the Definite Integral
Now we apply the limits from \( -\frac{\pi}{4} \) to \( \frac{\pi}{4} \):
\[ I = -\frac{1}{2} \left[ e^{-\frac{\pi}{4}} \left( \sin \frac{\pi}{4} + \cos \frac{\pi}{4} \right) - e^{\frac{\pi}{4}} \left( \sin \left(-\frac{\pi}{4}\right) + \cos \left(-\frac{\pi}{4}\right) \right) \right] \]
Since \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \):
\[ I = -\frac{1}{2} \left[ e^{-\frac{\pi}{4}} \left( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \right) - e^{\frac{\pi}{4}} \left( -\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \right) \right] \]
This simplifies to:
\[ I = -\frac{1}{2} \left[ e^{-\frac{\pi}{4}} \cdot \frac{2}{\sqrt{2}} + 0 \right] \]
Thus,
\[ I = -\frac{1}{\sqrt{2}} e^{-\frac{\pi}{4}} \]

### Final Result
The final answer is:
\[ I = -\frac{\sqrt{2}}{2} e^{-\frac{\pi}{4}} \]