Step by step
The question
Question: find the integral from 1 to 4 of 6/(2x + 1).
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.
With 2x + 1 at the bottom: ln of the bottom, divided by 2.
Its derivative would bring a 2 out, by the chain rule.
Careful!
The power rule fails for 1/x: the new power would be 0, and we cannot divide by 0.
Use ln.
2 Integrate
Divide by 2: 6/2 is 3.
No + C: we work out top minus bottom.
3 Top minus bottom
Put in 4, then take away the same with 1.
Two logs taken away make one log of a fraction.
So the integral is 3 ln 3.
A calculator gives about 3.296.
4 Check
Let's check with a calculator: it adds up 1000 thin strips.
That gives about 3.296, the same as our answer.
Right!
Let's look back.
First: the rules.
Then: integrate.
Then: top minus bottom.
Last: check.