Step by step
The question
Question: find the integral of e³ˣ.
∫ e3x 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 e³ˣ: its derivative is 3 times e³ˣ, by the chain rule.
That 3 must not be there.
So we divide by 3.
∫ e3x dx = 13e3x + C
Careful!
The derivative of e³ˣ is 3 times e³ˣ.
So e³ˣ alone is not the answer: divide by 3.
Careful: not ∫ e3x dx = e3x ✗
2 Integrate
Divide by 3: we get 1/3 times e³ˣ.
Add C: the slope cannot tell us a number added on the end.
So the answer is 1/3 × e³ˣ + C.
∫ e3x dx = 13e3x + C
3 Check
Let's check: take the derivative of our answer.
e to the kx gives k times e to the kx.
derivative: 13 × 3e3x
The numbers cancel, and we get e³ˣ: the question again.
Right!
= e3x
Let's look back.
First: the rules.
Then: integrate.
Last: check.
Answer
∫ e3x dx = 13e3x + C