Step-by-step explanation:
To find the integral of \( e^{-5x} \), we use the standard integration technique for exponential functions. Here's the step-by-step process:
\[ \int e^{-5x} \, dx \]
1. Recognize that \( e^{-5x} \) is an exponential function where the exponent is a linear function of \( x \).
2. Use the formula for integrating an exponential function of the form \( e^{ax} \):
\[
\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C
\]
where \( a \) is a constant and \( C \) is the constant of integration.
3. In this case, \( a = -5 \).
4. Substitute \( a \) with \(-5\) in the formula:
\[
\int e^{-5x} \, dx = \frac{1}{-5} e^{-5x} + C
\]
5. Simplify the result:
\[
\int e^{-5x} \, dx = -\frac{1}{5} e^{-5x} + C
\]
So, the integral of \( e^{-5x} \) with respect to \( x \) is:
\[
-\frac{1}{5} e^{-5x} + C
\]
