Step by step
The question
Question: find ∫ (8 x³ + 3 x² − 1) dx, the antiderivative of 8 x³ + 3 x² − 1.
1 The rule
The power rule takes 1 off the power, and the power comes down in front.
Backwards, we undo both: add 1 to the power, then divide by the new power.
∫ and dx just mean: find the function whose derivative this is.
Why + C?
A number alone has slope 0.
So any number could have been there.
We write C for it.
Careful!
Every antiderivative ends with + C.
Without it, the answer is not complete.
2 Each term
Use it on each term.
A number alone, like 5, comes from 5x: it gets an x.
Divide the numbers where we can: 8 divided by 4 is 2, 3 divided by 3 is 1.
So the answer is 2 x⁴ + x³ − x + C.
3 Check
Let's check: take the derivative of our answer.
The C gives 0.
We get 8 x³ + 3 x² − 1: the question again.
Right!
Let's look back.
First: the rule.
Then: each term.
Last: check.