Choose the correct Answer of the Following Questions : `int_0^(pi/4) e^x sinx dx =`

Options

A. 1
B. -1/2
C. 0
D. 1/2

Answer

To solve the definite integral \( \int_0^{\frac{\pi}{4}} e^x \sin x \, dx \), we can use the formula for the integral of the product of an exponential function and a sine function.

### Step-by-Step Solution:

1. **Identify the Integral**: We need to evaluate \( \int_0^{\frac{\pi}{4}} e^x \sin x \, dx \).

2. **Use the Formula**: The formula for the integral of \( e^{ax} \sin(bx) \) is:
\[ \int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} \left( a \sin(bx) - b \cos(bx) \right) + C \]
For our case, \( a = 1 \) and \( b = 1 \).

3. **Apply the Formula**: Substitute \( a = 1 \) and \( b = 1 \) into the formula:
\[ \int e^x \sin x \, dx = \frac{e^x}{1^2 + 1^2} \left( 1 \sin x - 1 \cos x \right) + C = \frac{e^x}{2} \left( \sin x - \cos x \right) + C \]

4. **Evaluate the Definite Integral**: Now we need to evaluate:
\[ \int_0^{\frac{\pi}{4}} e^x \sin x \, dx = \left[ \frac{e^x}{2} (\sin x - \cos x) \right]_0^{\frac{\pi}{4}} \]

5. **Calculate the Upper Limit**: First, evaluate at \( x = \frac{\pi}{4} \):
\[ = \frac{e^{\frac{\pi}{4}}}{2} \left( \sin \frac{\pi}{4} - \cos \frac{\pi}{4} \right) = \frac{e^{\frac{\pi}{4}}}{2} \left( \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} \right) = \frac{e^{\frac{\pi}{4}}}{2} \cdot 0 = 0 \]

6. **Calculate the Lower Limit**: Now evaluate at \( x = 0 \):
\[ = \frac{e^0}{2} \left( \sin 0 - \cos 0 \right) = \frac{1}{2} (0 - 1) = -\frac{1}{2} \]

7. **Combine the Results**: Now subtract the two results:
\[ \int_0^{\frac{\pi}{4}} e^x \sin x \, dx = 0 - \left(-\frac{1}{2}\right) = \frac{1}{2} \]

### Final Answer:
Thus, the value of the definite integral \( \int_0^{\frac{\pi}{4}} e^x \sin x \, dx \) is \( \frac{1}{2} \).