Step by step
The question
Question: find the integral of e⁵ˣ.
∫ e5x 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 e⁵ˣ: its derivative is 5 times e⁵ˣ, by the chain rule.
That 5 must not be there.
So we divide by 5.
∫ e5x dx = 15e5x + C
Careful!
The derivative of e⁵ˣ is 5 times e⁵ˣ.
So e⁵ˣ alone is not the answer: divide by 5.
Careful: not ∫ e5x dx = e5x ✗
2 Integrate
Divide by 5: we get 1/5 times e⁵ˣ.
Add C: the slope cannot tell us a number added on the end.
So the answer is 1/5 × e⁵ˣ + C.
∫ e5x dx = 15e5x + C
3 Check
Let's check: take the derivative of our answer.
e to the kx gives k times e to the kx.
derivative: 15 × 5e5x
The numbers cancel, and we get e⁵ˣ: the question again.
Right!
= e5x
Let's look back.
First: the rules.
Then: integrate.
Last: check.
Answer
∫ e5x dx = 15e5x + C