Integral of exp(3*x) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 3 x$.

      Then let $du = 3 dx$ and substitute $\frac{du}{3}$:

      $$\int \frac{e^{u}}{9}\, du$$

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

         $$\int \frac{e^{u}}{3}\, du = \frac{\int e^{u}\, du}{3}$$

         1. The integral of the exponential function is itself.

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

         So, the result is: $\frac{e^{u}}{3}$

      Now substitute $u$ back in:

      $$\frac{e^{3 x}}{3}$$

   Method #2

   1. Let $u = e^{3 x}$.

      Then let $du = 3 e^{3 x} dx$ and substitute $\frac{du}{3}$:

      $$\int \frac{1}{9}\, du$$

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

         $$\int \frac{1}{3}\, du = \frac{\int 1\, du}{3}$$

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

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

         So, the result is: $\frac{u}{3}$

      Now substitute $u$ back in:

      $$\frac{e^{3 x}}{3}$$

2. Add the constant of integration:

   $$\frac{e^{3 x}}{3}+ \mathrm{constant}$$

The answer is: