Step by step
The question
Question: find the integral from 0 to 2π of cos² x.
1 Use the half-angle rule
The half-angle rule: cos squared of x is 1 plus cos 2x, all over 2.
It changes a square into a plain cos.
Use it in the integral.
Split the fraction: 1/2, plus 1/2 cos 2x.
Careful!
The power rule does not work on cos squared: the derivative of cos x is not there.
Use the half-angle rule.
2 Integrate
cos 2x integrates to sin 2x/2: divide by the 2 inside.
No + C: top minus bottom.
3 Top minus bottom
sin of a whole number of halves of a turn is 0.
Put in 2π, then take away the same with 0.
The sin parts are 0.
So the integral is π, about 3.14.
4 Check
Let's check with a calculator: it adds up 1000 thin strips.
That gives about 3.14, the same as our answer.
Right!
Let's look back.
First: use the half-angle rule.
Then: integrate.
Then: top minus bottom.
Last: check.