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

Options

A. `(e^(pi//4))/(sqrt(2))`
B. `(1)/(2)`
C. `sqrt(2)e^(pi//4)`
D. `e^(pi//4)`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{4}} e^{x} \sin x \, dx \), we will use integration by parts.

### Step 1: Set up integration by parts
We will use the formula for integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
Let:
- \( u = \sin x \) (thus \( du = \cos x \, dx \))
- \( dv = e^{x} \, dx \) (thus \( v = e^{x} \))

### Step 2: Apply integration by parts
Now we apply the integration by parts formula:
\[ I = \left[ \sin x \cdot e^{x} \right]_{0}^{\frac{\pi}{4}} - \int_{0}^{\frac{\pi}{4}} e^{x} \cos x \, dx \]

### Step 3: Evaluate the boundary term
Now we evaluate the boundary term:
\[ \left[ \sin x \cdot e^{x} \right]_{0}^{\frac{\pi}{4}} = \sin\left(\frac{\pi}{4}\right) e^{\frac{\pi}{4}} - \sin(0) e^{0} \]
Calculating this gives:
\[ = \frac{1}{\sqrt{2}} e^{\frac{\pi}{4}} - 0 = \frac{1}{\sqrt{2}} e^{\frac{\pi}{4}} \]

### Step 4: Set up the new integral
Now we have:
\[ I = \frac{1}{\sqrt{2}} e^{\frac{\pi}{4}} - \int_{0}^{\frac{\pi}{4}} e^{x} \cos x \, dx \]
Let’s denote the new integral as \( J = \int_{0}^{\frac{\pi}{4}} e^{x} \cos x \, dx \).

### Step 5: Apply integration by parts to \( J \)
We will apply integration by parts again to \( J \):
Let:
- \( u = \cos x \) (thus \( du = -\sin x \, dx \))
- \( dv = e^{x} \, dx \) (thus \( v = e^{x} \))

Using the integration by parts formula:
\[ J = \left[ \cos x \cdot e^{x} \right]_{0}^{\frac{\pi}{4}} - \int_{0}^{\frac{\pi}{4}} e^{x} (-\sin x) \, dx \]
This simplifies to:
\[ J = \left[ \cos x \cdot e^{x} \right]_{0}^{\frac{\pi}{4}} + \int_{0}^{\frac{\pi}{4}} e^{x} \sin x \, dx \]
Evaluating the boundary term:
\[ \left[ \cos x \cdot e^{x} \right]_{0}^{\frac{\pi}{4}} = \cos\left(\frac{\pi}{4}\right) e^{\frac{\pi}{4}} - \cos(0) e^{0} = \frac{1}{\sqrt{2}} e^{\frac{\pi}{4}} - 1 \]
Thus, we have:
\[ J = \frac{1}{\sqrt{2}} e^{\frac{\pi}{4}} - 1 + I \]

### Step 6: Substitute back into the equation for \( I \)
Now we substitute \( J \) back into the equation for \( I \):
\[ I = \frac{1}{\sqrt{2}} e^{\frac{\pi}{4}} - \left( \frac{1}{\sqrt{2}} e^{\frac{\pi}{4}} - 1 + I \right) \]
This simplifies to:
\[ I = \frac{1}{\sqrt{2}} e^{\frac{\pi}{4}} - \frac{1}{\sqrt{2}} e^{\frac{\pi}{4}} + 1 - I \]
\[ 2I = 1 \implies I = \frac{1}{2} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{4}} e^{x} \sin x \, dx = \frac{1}{2} \]