Step by step
The question
Question: find the integral from −1 to 1 of x² + 2.
∫ from −1 to 1 of (x2 + 2) dx =
1 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
Now we have F(x) = x³ over 3 + 2x.
F(x) = x33 + 2x
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 −1 and 1 on the bar.
= (x33 + 2x) from −1 to 1
Put in 1 first, then take away the same with −1.
= (133 + 2(1)) − ((−1)33 + 2(−1))
Work out each bracket.
= 73 − (−73)
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(−1) − F(1) ✗
So the integral is 14/3.
= 143
3 Check
Let's check F: its derivative is x² + 2, the function we started with.
Right!
F′(x) = x2 + 2
Let's look back.
First: the antiderivative.
Then: top minus bottom.
Last: check.
Answer
= 143