Step by step
The question
Question: find the integral from 1 to 3 of 2 over x.
Two new rules: one for e to a power, one for 1 over x.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The rules
Now 1 over x.
The power rule cannot do it: x to the −1 would become x to the 0, over 0.
Instead: the derivative of ln x is 1 over x.
So the integral of 1/x is ln|x| + C.
The bars mean the size of x.
ln only takes positive numbers, and x may be negative.
Careful!
Not the power rule here: use ln.
2 Integrate
2 times 1 over the bottom gives 2 ln|x|.
No + C: we work out top minus bottom.
3 Top minus bottom
Put in 3, then take away the same with 1.
ln 1 is 0, because e to the 0 is 1.
So the integral is 2 ln 3.
A calculator gives about 2.197.
4 Check
Let's check with a calculator: it adds up 1000 thin strips.
That gives about 2.197, the same as our answer.
Right!
5 Pick the choice
Our answer is 2ln 3 ≈ 2.197.
That is choice D.
6 Why not the others
A is 1.099.
Wrong: lost the 2: 2/x gives 2ln x.
B is 1.333.
Wrong: used 2(1 − 1/3): the integral of 1/x is ln x.
C is 4.
Wrong: put 3 and 1 into 2x: the integral of 2/x is 2ln x.
Let's look back.
First: the rules.
Then: integrate.
Then: top minus bottom.
Then: check.
Then: pick the choice.
Last: why not the others.