Integral of e^(2x-y)dy dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 2 x - y$.

      Then let $du = - dy$ 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:

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

   Method #2

   1. Rewrite the integrand:

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

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

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

      1. Let $u = - y$.

         Then let $du = - dy$ 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:

         $$- e^{- y}$$

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

   Method #3

   1. Rewrite the integrand:

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

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

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

      1. Let $u = - y$.

         Then let $du = - dy$ 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:

         $$- e^{- y}$$

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

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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