`int e^(2x) sin x dx`

Answer

To solve the integral \( \int e^{2x} \sin x \, dx \), we can use the method of integration by parts or apply a standard result for integrals of the form \( \int e^{ax} \sin(bx) \, dx \). Here, we will use the standard result approach.

### Step-by-Step Solution:

1. **Identify Parameters**:
We have \( a = 2 \) and \( b = 1 \) from the integral \( \int e^{2x} \sin x \, dx \).

2. **Use the Standard Result**:
The standard result for the integral \( \int e^{ax} \sin(bx) \, dx \) is given by:
\[ \int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C \]
Substituting \( a = 2 \) and \( b = 1 \) into the formula, we get:
\[ a^2 + b^2 = 2^2 + 1^2 = 4 + 1 = 5 \]

3. **Substitute into the Formula**:
Now substituting \( a \) and \( b \) into the formula:
\[ \int e^{2x} \sin x \, dx = \frac{e^{2x}}{5} (2 \sin x - 1 \cos x) + C \]

4. **Simplify the Expression**:
This simplifies to:
\[ \int e^{2x} \sin x \, dx = \frac{e^{2x}}{5} (2 \sin x - \cos x) + C \]

### Final Answer:
Thus, the final result is:
\[ \int e^{2x} \sin x \, dx = \frac{e^{2x}}{5} (2 \sin x - \cos x) + C \]