Question

What is the antiderivative of `e^-x`?

Answer 1

`-e^(-x)+C`

Explanation:

The problem, written out, is:

`inte^(-x)dx`

The following rule will be used:

`inte^udu=e^u+C`

Using substitution:
Set `u=-x`, so we know `(du)/dx=-1` and `-du=dx`.

This gives us a simplified integral of

`=inte^u(-du)`

We can bring the negative sign out:

`=-inte^udu=-e^u+C=-e^(-x)+C`

Using intuition:

We know that the derivative of `e^(-x)` is `-e^(-x)`, but we want a derivative of the positive version `e^(-x)`. So, we make the antiderivative negative, causing the derivative of be positive:

`d/dx-e^(-x)=e^(-x)`

Then, we simply add `C`, the constant of integration.