Question

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

Answer 1

`-e^-x+C`

Explanation:

We will use the following integral rule:

`inte^udu=e^u+C`

Thus, to integrate

`inte^-xdx`

We set `u=-x`. Thus, we have:

`color(red)(u=-x)" "=>" "(du)/dx=-1" "=>" "color(blue)(du=-dx)`

Since we have only `dx` in the integral and not `-dx`, multiply the interior of the integral by `-1`. Balance this by multiplying the exterior by `-1` as well.

`inte^-xdx=-inte^color(red)(-x)color(blue)((-1)dx)=-inte^color(red)ucolor(blue)(du)`

This is the rule we knew originally. Don't forget that the integral is multiplied by `-1`:

`-inte^udu=-e^u+C=barul|color(white)(a/a)-e^-x+Ccolor(white)(a/a)|`