Step by step
The question
Question: find the integral of 3e²ˣ.
∫ 3e2x 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 e²ˣ: its derivative is 2 times e²ˣ, by the chain rule.
That 2 must not be there.
So we divide by 2.
∫ e2x dx = 12e2x + C
Careful!
The derivative of e²ˣ is 2 times e²ˣ.
So e²ˣ alone is not the answer: divide by 2.
Careful: not ∫ e2x dx = e2x ✗
2 Integrate
Divide by 2: 3/2 is 3/2.
Add C: the slope cannot tell us a number added on the end.
So the answer is 3/2 × e²ˣ + C.
∫ 3e2x dx = 32e2x + C
3 Check
Let's check: take the derivative of our answer.
e to the kx gives k times e to the kx.
derivative: 32 × 2e2x
The numbers cancel, and we get 3e²ˣ: the question again.
Right!
= 3e2x
Let's look back.
First: the rules.
Then: integrate.
Last: check.
Answer
∫ 3e2x dx = 32e2x + C