Worked examples
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.
▶ Watch it solved
The question
Question: find ∫ (8 x³ + 3 x² − 1) dx, the antiderivative of 8 x³ + 3 x² − 1.
∫ (8x3 + 3x2 − 1) 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 alone, like 5, comes from 5x: it gets an x.
= 8 × x44 + 3 × x33 − x + C
Divide the numbers where we can: 8 divided by 4 is 2, 3 divided by 3 is 1.
8 ÷ 4 = 2, 3 ÷ 3 = 1
So the answer is 2 x⁴ + x³ − x + C.
= 2x4 + 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!
(2x4 + x3 − x + C)′ = 8x3 + 3x2 − 1
Let's look back.
First: the rule.
Then: each term.
Last: check.
Answer = 2x4 + x3 − x + C
▶ Watch it solved
The question
Question: find ∫ 6/x³ dx, the antiderivative of 6/x³.
∫ 6x3 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 Write as powers
To use the rule, every x needs a power.
x at the bottom is a minus power on top; a root is the power one half.
= ∫ 6x−3 dx
3 Each term
Use it on each term.
A number in front just waits.
= 6 × x−2−2 + C
Add 1 to the power: −3 + 1 is −2.
−3 + 1 = −2
So the answer is −3/x² + C.
= −3x2 + C
4 Check
Let's check: take the derivative of our answer.
The C gives 0.
We get 6 x⁻³: the question again.
Right!
(−3x2 + C)′ = 6x−3
Let's look back.
First: the rule.
Then: write as powers.
Then: each term.
Last: check.
▶ Watch it solved
