Worked example

∫₀^π cos²x dx

Steps

1. cos²x = (1 + cos 2x)/2
   Power-reducing formula. · Power-reducing formula for cos²
2. ∫₀^π = [x/2 + sin 2x/4]₀^π
   Integrate.
3. = π/2 + 0 − 0
   sin 2π = sin 0 = 0.
4. = π/2
   The average value of cos² over a half-period is 1/2 — a useful sanity check.

Identities used

• Power-reducing formula for cos² — The cosine twin: ∫cos²x dx, RMS values of a wave, and any average of cos² over a whole number of periods (the cosine term integrates to 0 and 1/2 is left).

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.

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