Worked example

∫ sin³x dx

Steps

1. sin³x = sin x(1 − cos²x)
   Peel one sine off for the substitution — the standard odd-power trick. · Pythagorean identity
2. u = cos x, du = −sin x dx
   Substitution.
3. ∫ = −∫(1 − u²)du = −u + u³/3 + C
   Integrate the polynomial.
4. = −cos x + cos³x/3 + C
   Back-substitute.
5. Alternative: sin³x = (3 sin x − sin 3x)/4
   Reduction formula, then integrate three simple terms. · Reduction formula for sin³

Identities used

• Reduction formula for sin³ — Cubic powers: an integral of sin³x, or a Fourier-style rewrite. Odd powers reduce to a sum of sines of x, 3x, 5x … with binomial-looking coefficients.
• 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.

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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