Step by step
The question
Question: find the integral from 1 to 3 of 2x + 1.
∫ from 1 to 3 of (2x + 1) dx =
1 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
F(x) = 2 × x22 + x
Now we have F(x) = x² + x.
= x2 + x
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 3 on the bar.
= (x2 + x) from 1 to 3
Put in 3 first, then take away the same with 1.
= (32 + 3) − (12 + 1)
Work out each bracket.
= 12 − 2
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(1) − F(3) ✗
So the integral is 10.
= 10
3 Check
Let's check with geometry.
Under a straight line the shape is a trapezoid: heights 3 and 7, width 2.
Its area is 10.
The same!
Right!
trapezoid: 3 + 72 × 2 = 10
Let's look back.
First: the antiderivative.
Then: top minus bottom.
Last: check.
Answer
= 10