Step by step
The question
Question: find the integral of e⁻ˣ.
∫ e−x dx =
1 The rules
First, e to the x.
Its derivative is eˣ itself: its slope equals its height.
So its integral is eˣ + C too.
∫ ex dx = ex + C
With e⁻ˣ: its derivative is −1 times e⁻ˣ, by the chain rule.
That −1 must not be there.
So we divide by −1.
∫ e−x dx = 1−1e−x + C
Careful!
The derivative of e⁻ˣ is −1 times e⁻ˣ.
So e⁻ˣ alone is not the answer: divide by −1.
Careful: not ∫ e−x dx = e−x ✗
2 Integrate
Divide by −1: we get −1 times e⁻ˣ.
Add C: the slope cannot tell us a number added on the end.
So the answer is −e⁻ˣ + C.
∫ e−x dx = −e−x + C
3 Check
Let's check: take the derivative of our answer.
e to the kx gives k times e to the kx.
derivative: −−1e−x
The numbers cancel, and we get e⁻ˣ: the question again.
Right!
= e−x
Let's look back.
First: the rules.
Then: integrate.
Last: check.
Answer
∫ e−x dx = −e−x + C