Power Reduction Arsenal

Calculus 2 · Axiom Academy

The trig identities that turn impossible-looking integrals into easy ones — by trading powers and products for plain sums of sines and cosines. The whole trick: kill the power A squared sine or cosine has no clean antiderivative on its own — you can't just integrate the way you'd integrate . The power-reduction identities are the way out: they rewrite a squared trig function as a constant plus a single first-power cosine, and a first-power cosine integrates in one step. Watch get rewritten as . The wavy squared curve is exactly the same as a flat baseline at minus a single cosine wave — and that flat-plus-cosine form is trivial to integrate. The running average climbs to the level the identity predicts: . Why even powers reduce: it's all about the average Drag the slider to slide a window across the curve and watch the running average settle. Both and wobble around the same level — — which is exactly the constant the power-reduction identity pulls out front. Flip between them; the wave changes, the average doesn't. That constant is the whole payoff: — no power left to fight. Products of different angles: split them into sums A product like has no obvious antiderivative — but the product-to-sum identity rewrites it as , two plain sines you integrate term by term. Drag the two frequencies and watch the beating product (top) become the sum of its two clean waves (bottom). Pattern recognition: which identity fits

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