Integral of e^(2-x) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 2 - x$.

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

      $$\int \left(- e^{u}\right)\, du$$

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

         $$\text{False}$$

         1. The integral of the exponential function is itself.

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

         So, the result is: $- e^{u}$

      Now substitute $u$ back in:

      $$- e^{2 - x}$$

   Method #2

   1. Rewrite the integrand:

      $$e^{2 - x} = e^{2} e^{- x}$$

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

      $$\int e^{2} e^{- x}\, dx = e^{2} \int e^{- x}\, dx$$

      1. Let $u = - x$.

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

         $$\int \left(- e^{u}\right)\, du$$

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

            $$\text{False}$$

            1. The integral of the exponential function is itself.

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

            So, the result is: $- e^{u}$

         Now substitute $u$ back in:

         $$- e^{- x}$$

      So, the result is: $- e^{2} e^{- x}$

   Method #3

   1. Rewrite the integrand:

      $$e^{2 - x} = e^{2} e^{- x}$$

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

      $$\int e^{2} e^{- x}\, dx = e^{2} \int e^{- x}\, dx$$

      1. Let $u = - x$.

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

         $$\int \left(- e^{u}\right)\, du$$

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

            $$\text{False}$$

            1. The integral of the exponential function is itself.

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

            So, the result is: $- e^{u}$

         Now substitute $u$ back in:

         $$- e^{- x}$$

      So, the result is: $- e^{2} e^{- x}$

2. Add the constant of integration:

   $$- e^{2 - x}+ \mathrm{constant}$$

The answer is:

$$- e^{2 - x}+ \mathrm{constant}$$