Worked example

∫ sin x cos x dx — two ways

Answer

sin²x/2 + C = −cos 2x/4 + C (the forms differ by a constant)

Substitution u = sin x, du = cos x dx
First route: ∫u du = u²/2 = sin²x/2 + C.
sin x cos x = (1/2) sin 2x
Second route: double-angle formula rearranged. · Double-angle identity for sine
∫ = −cos 2x/4 + C
Integrate the sine of a double angle.
Both are right
sin²x/2 = (1 − cos 2x)/4, and 1/4 is absorbed into C — worth seeing, because students lose marks believing one route is an error.

sin 2x = 2 sin x cos x

Double-angle identity for sine — The question is in 2x and the answer (or the next step) is in x — or you need to integrate/solve something containing sin x cos x, which is this formula run backwards.

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.