Step by step
The question
Question: find the integral of cos² x.
∫ cos2 x dx =
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.
cos2 x = 1 + cos 2x2
Use it in the integral.
= ∫ 1 + cos 2x2 dx
Split the fraction: 1/2, plus 1/2 cos 2x.
= ∫ (12 + 12 cos 2x) dx
Careful!
The power rule does not work on cos squared: the derivative of cos x is not there.
Use the half-angle rule.
Careful: not ∫ cos2 x dx = cos3 x3 ✗
2 Integrate
cos 2x integrates to sin 2x/2: divide by the 2 inside.
∫ cos 2x dx = sin 2x2
Put the pieces together and add C: 1/2 × x + sin 2x/4 + C.
= 12x + sin 2x4 + C
3 Check
Let's check: the slope of our answer F near x = 1, with a calculator in radians: about 0.292.
F(1.001) − F(0.999)0.002 ≈ 0.292
The question at x = 1 gives 0.292.
The same: the slope of the answer is the question.
Right!
cos² 1 ≈ 0.292
Let's look back.
First: use the half-angle rule.
Then: integrate.
Last: check.
Answer
= 12x + sin 2x4 + C