Integral of 2exp(2x) dx

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

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

   1. There are multiple ways to do this integral.

      Method #1

      1. Let $u = 2 x$.

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

         $$\int \frac{e^{u}}{4}\, du$$

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

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

            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:

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

      Method #2

      1. Let $u = e^{2 x}$.

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

         $$\int \frac{1}{4}\, du$$

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

            $$\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}$$

            1. The integral of a constant is the constant times the variable of integration:

               $$\int 1\, du = u$$

            So, the result is: $\frac{u}{2}$

         Now substitute $u$ back in:

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

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

2. Add the constant of integration:

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

The answer is:

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