Step by step
The question
Question: find ∫ (5 x⁴ − 6x + 7) dx, the antiderivative of 5 x⁴ − 6x + 7.
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 alone, like 5, comes from 5x: it gets an x.
Divide the numbers where we can: 5 divided by 5 is 1, 6 divided by 2 is 3.
So the answer is x⁵ − 3 x² + 7x + C.
3 Check
Let's check: take the derivative of our answer.
The C gives 0.
We get 5 x⁴ − 6x + 7: the question again.
Right!
Let's look back.
First: the rule.
Then: each term.
Last: check.