Step by step
The question
Question: find the integral from 0 to 2 of x³ − x.
∫ from 0 to 2 of (x3 − x) dx =
1 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
Now we have F(x) = x⁴ over 4 − x² over 2.
F(x) = x44 − x22
No + C here: in F(b) minus F(a), the two Cs cancel.
no + C here: it cancels
2 Top minus bottom
The rule: put in the top number, then take away the bottom number.
∫ from a to b of f(x) dx = F(b) − F(a)
Write F(x) with 0 and 2 on the bar.
= (x44 − x22) from 0 to 2
Put in 2 first, then take away the same with 0.
= (244 − 222) − (044 − 022)
Work out each bracket.
= 2 − 0
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(0) − F(2) ✗
So the integral is 2.
= 2
3 Check
Let's check F: its derivative is x³ − x, the function we started with.
Right!
F′(x) = x3 − x
Let's look back.
First: the antiderivative.
Then: top minus bottom.
Last: check.
Answer
= 2