Find `int(e^(-5x)+3)dx.`

Options

A. `-(e^(-5))/(5)+3`
B. `-(e^(-5))/(5)+3x`
C. `(e^(-5))/(5)+3x`
D. `-(e^(-5))/(5)+0`

Explanation

To solve the integral \( \int (e^{-5x} + 3) \, dx \), we can break it down into two separate integrals:

1. **Separate the integral**:
\[ \int (e^{-5x} + 3) \, dx = \int e^{-5x} \, dx + \int 3 \, dx \]

2. **Integrate \( e^{-5x} \)**:
To integrate \( e^{-5x} \), we use the formula for integrating exponential functions:
\[ \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \]
Here, \( a = -5 \). Thus,
\[ \int e^{-5x} \, dx = \frac{1}{-5} e^{-5x} + C_1 = -\frac{1}{5} e^{-5x} + C_1 \]

3. **Integrate \( 3 \)**:
The integral of a constant is simply the constant multiplied by \( x \):
\[ \int 3 \, dx = 3x + C_2 \]

4. **Combine the results**:
Now, we combine the results of both integrals:
\[ \int (e^{-5x} + 3) \, dx = -\frac{1}{5} e^{-5x} + 3x + C \]
where \( C = C_1 + C_2 \) is the constant of integration.

Thus, the final answer is:
\[ \int (e^{-5x} + 3) \, dx = -\frac{1}{5} e^{-5x} + 3x + C \]