Step by step
The question
Question: find the integral of sin² x.
∫ sin2 x dx =
1 Use the half-angle rule
The half-angle rule: sin squared of x is 1 minus cos 2x, all over 2.
It changes a square into a plain cos.
sin2 x = 1 − cos 2x2
Use it in the integral.
= ∫ 1 − cos 2x2 dx
Split the fraction: 1/2, minus 1/2 cos 2x.
= ∫ (12 − 12 cos 2x) dx
Careful!
The power rule does not work on sin squared: the derivative of sin x is not there.
Use the half-angle rule.
Careful: not ∫ sin2 x dx = sin3 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 0.708.
F(1.001) − F(0.999)0.002 ≈ 0.708
The question at x = 1 gives 0.708.
The same: the slope of the answer is the question.
Right!
sin² 1 ≈ 0.708
Let's look back.
First: use the half-angle rule.
Then: integrate.
Last: check.
Answer
= 12x − sin 2x4 + C