Step by step
The question
Question: find ∫ 3 x² dx, the antiderivative of 3 x².
∫ 3x2 dx =
1 The rule
The power rule takes 1 off the power, and the power comes down in front.
x3 → 3x2 (derivative)
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.
∫ xn dx = xn + 1n + 1 + C
Why + C?
A number alone has slope 0.
So any number could have been there.
We write C for it.
+ C: any number; its derivative is 0
Careful!
Every antiderivative ends with + C.
Without it, the answer is not complete.
Careful: never forget + C
2 Each term
Use it on each term.
A number in front just waits.
= 3 × x33 + C
Divide the numbers where we can: 3 divided by 3 is 1.
3 ÷ 3 = 1
So the answer is x³ + C.
= x3 + C
3 Check
Let's check: take the derivative of our answer.
The C gives 0.
We get 3 x²: the question again.
Right!
(x3 + C)′ = 3x2
Let's look back.
First: the rule.
Then: each term.
Last: check.
Answer
= x3 + C