Question

What is the antiderivative of `e^(2x)`?

Answer 1

Antiderivative is another name for the Integral( if by some misfortune you didnt know)
So,

`inte^(2x) = 1l/2int 2e^(2x) dx`

You can see that `2dx = d(2x)`

that is `2` is the derivative of `2x`

It follows : `1/2int e^(2x) d(2x)`
NOTE: this is the same as letting `u = 2x`

`1/2int e^u du = 1/2e^u`
`= 1/2e^(2x)`

Generally, `int e^(ax) = 1/ae^(ax)`

Answer 2

It is `1/2 e^(2x)`.

You can certainly use the technique of integration by substitution (reversing the chain rule) to find this, you can also reason as follows:

The antiderivative of `e^(2x)` is a function whose derivative is `e^(2x)`.

But we know some things about derivatives at this point of the course. Among other things, we know that the derivative of `e` to a power is `e` to the power times the derivative of the power.

So we know that the drivative of `e^(2x)` is `e^(2x)*2`. That's twice a big as what we want.

We also know that constant factors just hang out in front when we take derivatives, so if we stick a `1/2` out front, it will be there after we differentiate and we can cancel the two.

`f(x)=1/2e^(2x)` has `f'(x)=e^(2x)` so it is an antiderivative. The general antiderivative then is `1/2 e^(2x) +C`

Note
An important consequence of the Mean Value Theorem is that a function whose derivative is `0` is a constant function. And an immediate consequence of that is that if two functions have the same derivative, then they differ by a constant.
Therefore, any function that has derivative `e^(2x)` can ultimately be written as `1/2 e^(2x)+C` for some constant C.