Step by step
The question
Question: find the integral from 2 to 5 of x² − 9.
∫ from 2 to 5 of (x2 − 9) dx
An integral from a to b gives one number.
1 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
Now we have F(x) = x³ over 3 − 9x.
F(x) = x33 − 9x
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 2 and 5 on the bar.
= (x33 − 9x) from 2 to 5
Put in 5 first, then take away the same with 2.
= (533 − 9(5)) − (233 − 9(2))
First the powers and the times.
= (1253 − 45) − (83 − 18)
Then each bracket: the first is −10/3, the second is −46/3.
= −103 − (−463)
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(2) − F(5) ✗
So the integral is 12.
= 12
3 Check
Let's check F: its derivative is x² − 9, the function we started with.
Right!
F′(x) = x2 − 9
Let's look back.
First: the antiderivative.
Then: top minus bottom.
Last: check.
Answer
= 12