Trigonometric Integrals

Real Analysis · Axiom Academy

LESSON Trigonometric Integrals Strategies for integrating powers and products of sine, cosine, tangent, and secant — pick the move that matches the form. 1. Odd Powers — Peel One Factor, Convert the Rest For with at least one odd exponent, save a single factor of the odd function to become du , rewrite everything left over with , and substitute. The integral collapses into a polynomial in u . The sine power is odd, so peel off one and convert : Let , so — the saved factor becomes du : 2. Even Powers — Power-Reduction Formulas When every power is even there is no factor to peel off — instead, lower the power with the half-angle identities. Each squared term becomes a constant plus a single first-power ripple, which you can integrate term by term. Drop the power, then integrate the two simple pieces: For the engine is , and the derivative pair , . Save whichever derivative block is available, convert the rest with the identity, and substitute. The secant power is even, so split , save one for du , and convert the other with : With , , this is a polynomial integral: 4. Products of Different Angles — Product-to-Sum For with , none of the earlier moves apply — the angles differ. The product-to-sum identity rewrites the product as the average of two plain sine waves , each of which integrates immediately. Here a = 3 , b = 1 , so the product splits into waves of angle 4x and 2x : Integrate each sine on its own:

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