Question

How do you integrate `e^(4x) dx`?

Answer 1

`1/4e^(4x)+C`

Explanation:

We will use the integration rule for `e^x`:

`inte^udu=e^u+C`

So, for the given integral, let `u=4x`. This implies that `du=4dx`.

`inte^(4x)dx=1/4inte^(4x)*4dx=1/4inte^udu=1/4e^u+C`

Since `u=4x`:

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

We can differentiate this answer to check that we get `e^(4x)`. Indeed, through the chain rule, the `1/4` we had to add gets "undone" by the `4` coming from the power of `4x` via the chain rule.