Worked example
Simplify √(1 − cos x) so it can be integrated
Answer
√2·|sin(x/2)| — on 0 < x < 2π the absolute value is sin(x/2)
Steps
- 1 − cos x = 2sin²(x/2)Rearrange the half-angle identity. · Half-angle identity for sine
- √(1 − cos x) = √(2sin²(x/2)) = √2·|sin(x/2)|√(u²) = |u|, not u — the absolute value is the trap in this problem.
- For 0 < x < 2π, x/2 ∈ (0, π) so sin(x/2) > 0Drop the absolute value on that interval.
- = √2 sin(x/2)Now integrable: ∫ = −2√2 cos(x/2) + C.
Identities used
sin(x/2) = ±√((1 − cos x)/2)
- Half-angle identity for sine — A half angle appears, or an integral contains √(1 − cos x). Always decide the sign from the quadrant of x/2 BEFORE writing the root — that decision is the whole problem.
Check it yourself
Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.