Show step by step integration of exp(-x by a)

Certainly! To integrate the function $e^{-\frac{x}{a}}$ with respect to $x$, where $a$ is a constant, you can follow these steps:

1. Recognize the Integral: Identify the function to integrate: $\int e^{-\frac{x}{a}} \, dx$.

2. Use the Power Rule of Integration: Recall the power rule of integration: $\int x^n \, dx = \frac{1}{n+1} x^{n+1} + C$, where $n$ is any constant except -1.

3. Adjust the Exponential Function: The given function is in the form $e^{-\frac{x}{a}}$, which is similar to $e^{kx}$ where $k = -\frac{1}{a}$. The integration of $e^{kx}$ involves multiplying by the reciprocal of the coefficient of $x$ (which is $k$ in this case).

4. Apply the Integration Rule: Apply the integration rule to find the antiderivative of $e^{-\frac{x}{a}}$:

   $$\int e^{-\frac{x}{a}} \, dx = -a e^{-\frac{x}{a}} + C$$

   where $C$ is the constant of integration.

5. Final Answer: The final result of the integration is $-a e^{-\frac{x}{a}} + C$.

So, the step-by-step integration of $e^{-\frac{x}{a}}$ with respect to $x$ is:

$$\int e^{-\frac{x}{a}} \, dx = -a e^{-\frac{x}{a}} + C$$

where $C$ is the constant of integration.