Integral of e^(x-20) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = x - 20$.

      Then let $du = dx$ and substitute $du$:

      $$\int e^{u}\, du$$

      1. The integral of the exponential function is itself.

         $$\int e^{u}\, du = e^{u}$$

      Now substitute $u$ back in:

      $$e^{x - 20}$$

   Method #2

   1. Rewrite the integrand:

      $$e^{x - 20} = \frac{e^{x}}{e^{20}}$$

   2. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \frac{e^{x}}{e^{20}}\, dx = \frac{\int e^{x}\, dx}{e^{20}}$$

      1. The integral of the exponential function is itself.

         $$\int e^{x}\, dx = e^{x}$$

      So, the result is: $\frac{e^{x}}{e^{20}}$

2. Now simplify:

   $$e^{x - 20}$$

3. Add the constant of integration:

   $$e^{x - 20}+ \mathrm{constant}$$

The answer is:

$$e^{x - 20}+ \mathrm{constant}$$