Worked example

∫ sin²x dx

Answer

x/2 − sin 2x/4 + C

sin²x
A squared trig term cannot be integrated by substitution — reduce the power.
= (1 − cos 2x)/2
Power-reducing formula. · Power-reducing formula for sin²
∫ = x/2 − (sin 2x)/4 + C
Integrate the constant and the cosine.
check: d/dx[x/2 − sin 2x/4] = 1/2 − cos 2x/2 = sin²x
Differentiate back using the same formula in reverse.

sin²x = (1 − cos 2x)/2

Power-reducing formula for sin² — Almost always integration: ∫sin²x dx is not reachable as a power, but (1 − cos 2x)/2 integrates in two terms. Also for averaging sin² over a period.

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.