Step by step
The question
Question: find the integral of e⁻⁴ˣ.
∫ e−4x 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⁻⁴ˣ: its derivative is −4 times e⁻⁴ˣ, by the chain rule.
That −4 must not be there.
So we divide by −4.
∫ e−4x dx = 1−4e−4x + C
Careful!
The derivative of e⁻⁴ˣ is −4 times e⁻⁴ˣ.
So e⁻⁴ˣ alone is not the answer: divide by −4.
Careful: not ∫ e−4x dx = e−4x ✗
2 Integrate
Divide by −4: we get −1/4 times e⁻⁴ˣ.
Add C: the slope cannot tell us a number added on the end.
So the answer is −1/4 × e⁻⁴ˣ + C.
∫ e−4x dx = −14e−4x + C
3 Check
Let's check: take the derivative of our answer.
e to the kx gives k times e to the kx.
derivative: −14 × −4e−4x
The numbers cancel, and we get e⁻⁴ˣ: the question again.
Right!
= e−4x
Let's look back.
First: the rules.
Then: integrate.
Last: check.
Answer
∫ e−4x dx = −14e−4x + C