Step by step
The question
Question: find ∫ (12 x³ − 9 x² ) dx, the antiderivative of 12 x³ − 9 x².
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 in front just waits.
Divide the numbers where we can: 12 divided by 4 is 3, 9 divided by 3 is 3.
So the answer is 3 x⁴ − 3 x³ + C.
3 Check
Let's check: take the derivative of our answer.
The C gives 0.
We get 12 x³ − 9 x²: the question again.
Right!
Let's look back.
First: the rule.
Then: each term.
Last: check.