Question

How do you evaluate the integral `inte^(4x) dx`?

Answer 1

We will use `u`-substitution, letting `u = 4x`.

Thus, `du = 4dx`.

Also, we will use the constant law of integration, namely `int C*f(x)dx = C*int f(x) dx` to rewrite the integral so that it contains `du`:

`int e^(4x)dx = 1/4 int 4*e^(4x)dx`

Now, we will rewrite in terms of `u`:

`int e^(4x)dx = 1/4 int e^(u)du`

We know that the integral of `e^u du` will simply be `e^u`. Remember the constant of integration:

`int e^(4x)dx = 1/4 e^(u) + C`

Substituting back `u` gives:

`int e^(4x)dx = 1/4 e^(4x) + C`