Step by step
The question
Question: find ∫ (2 x⁵ − x² ) dx, the antiderivative of 2 x⁵ − x².
∫ (2x5 − x2) 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.
= 2 × x66 − x33 + C
So the answer is x⁶ over 3 − x³ over 3 + C.
= x63 − x33 + C
3 Check
Let's check: take the derivative of our answer.
The C gives 0.
We get 2 x⁵ − x²: the question again.
Right!
(x63 − x33 + C)′ = 2x5 − x2
Let's look back.
First: the rule.
Then: each term.
Last: check.
Answer
= x63 − x33 + C