Step by step
The question
Question: find the integral from −2 to 1 of 3x² + 2x.
∫ from −2 to 1 of (3x2 + 2x) dx =
1 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
F(x) = 3 × x33 + 2 × x22
Now we have F(x) = x³ + x².
= x3 + x2
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 1 on the bar.
= (x3 + x2) from −2 to 1
Put in 1 first, then take away the same with −2.
= (13 + 12) − ((−2)3 + (−2)2)
Work out each bracket.
= 2 − (−4)
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(−2) − F(1) ✗
So the integral is 6.
= 6
3 Check
Let's check F: its derivative is 3 x² + 2x, the function we started with.
Right!
F′(x) = 3x2 + 2x
Let's look back.
First: the antiderivative.
Then: top minus bottom.
Last: check.
Answer
= 6