Integral of e^(2x-y) dx

1. There are multiple ways to do this integral.

   Method #1

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

      Then let $du = 2 dx$ and substitute $\frac{du}{2}$:

      $$\int \frac{e^{u}}{2}\, du$$

      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: $\frac{e^{u}}{2}$

      Now substitute $u$ back in:

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

   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:

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

      1. Let $u = 2 x$.

         Then let $du = 2 dx$ and substitute $\frac{du}{2}$:

         $$\int \frac{e^{u}}{2}\, du$$

         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: $\frac{e^{u}}{2}$

         Now substitute $u$ back in:

         $$\frac{e^{2 x}}{2}$$

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

   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:

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

      1. Let $u = 2 x$.

         Then let $du = 2 dx$ and substitute $\frac{du}{2}$:

         $$\int \frac{e^{u}}{2}\, du$$

         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: $\frac{e^{u}}{2}$

         Now substitute $u$ back in:

         $$\frac{e^{2 x}}{2}$$

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

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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