Lecture 7.2 - Trigonometric Integrals.pdf - MATH-2212...

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MATH-2212: Calculus II7.2. Trigonometric Integrals7.2. Trigonometric Integrals0.Review of Some Trigonometric IdentitiesTrigonometric Pythagorean Identitiessin2θ+ cos2θ= 1tan2θ+ 1 = sec2θcot2θ+ 1 = csc2θDouble-Angle Identitiessin 2θ= 2 sinθcosθcos 2θ= cos2θ-sin2θ= 2 cos2θ-1 = 1-2 sin2θPower Reducing Identities, a.k.a. Half-Angle Identitiessin2θ=12(1-cos 2θ)cos2θ=12(1 + cos 2θ)Sum and Difference Identitiessin(α+β) = sinαcosβ+ cosαsinβcos(α+β) = cosαcosβ-sinαsinβsin(α-β) = sinαcosβ-cosαsinβcos(α-β) = cosαcosβ+ sinαsinβProduct-to-Sum Identitiessinαcosβ=12[sin(α+β) + sin(α-β)]cosαcosβ=12[cos(α+β) + cos(α-β)]sinαsinβ=12[cos(α-β)-cos(α+β)]1.Products ofsinandcosFunctionsHere we’re talking about integrals of the formZsinm(ax) cosn(ax)dx, where the exponentsmandncan be pretty much arbitrary, but the arguments of the sine and cosine functions must be thesame.– 1–
MATH-2212: Calculus II7.2. Trigonometric IntegralsThe Rulefor integrating expressions of the formZsinm(ax) cosn(ax

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