Step by step
The question
Question: find ∫ 6x dx, the antiderivative of 6x.
∫ 6x 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.
= 6 × x22 + C
Divide the numbers where we can: 6 divided by 2 is 3.
6 ÷ 2 = 3
So the answer is 3 x² + C.
= 3x2 + C
3 Check
Let's check: take the derivative of our answer.
The C gives 0.
We get 6x: the question again.
Right!
(3x2 + C)′ = 6x
Let's look back.
First: the rule.
Then: each term.
Last: check.
Answer
= 3x2 + C