Worked example

∫ cos³x dx

Answer

sin x − sin³x/3 + C

cos³x = cos x(1 − sin²x)
Odd power again: peel one cosine. · Pythagorean identity
u = sin x
Substitution, du = cos x dx.
∫ = u − u³/3 + C = sin x − sin³x/3 + C
Integrate and back-substitute.
Reduction route: cos³x = (3cos x + cos 3x)/4
Gives (3sin x + sin 3x)/4 + C — same function up to a constant. · Reduction formula for cos³

cos³x = (3 cos x + cos 3x)/4

sin²θ + cos²θ = 1

Reduction formula for cos³ — Cubic cosine — the sign is '+' here where the sine version is '−'. Used in integrals of cos³x and when analysing the third harmonic.
Pythagorean identity — Two squared trig terms of the same angle are added: the pair is exactly 1, so replace both with 1 — or, going the other way, replace a stubborn 1 with sin²θ + cos²θ to get a common denominator.

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.