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NSWMaths Extension 2Quick questions

Further Integration (MEX-C1)

Quick questions on Trigonometric integrals in HSC Mathematics Extension 2: powers of sine and cosine (odd and even strategies), powers of tangent and secant, the integral of sec x and sec cubed x, and products of sines and cosines via product-to-sum identities

3short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is show the structural step for the mark?
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Write the line that exposes the method: the peeled factor with the Pythagorean conversion (cos⁡3x=cos⁡x(1−sin⁡2x)\cos^3 x = \cos x(1-\sin^2 x)), the reserved sec⁡2x\sec^2 x or sec⁡xtan⁡x\sec x\tan x, or the product-to-sum line. That line is usually the first mark; jumping to a uu-substitution without it can be marked down.
What is the wrong reserved factor for tan and sec?
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Even power of sec⁡\sec reserves sec⁡2x\sec^2 x with u=tan⁡xu = \tan x; odd power of tan⁡\tan reserves sec⁡xtan⁡x\sec x\tan x with u=sec⁡xu = \sec x. Reserving the wrong one leaves a factor that is not the right dudu.
What is sign in the sin⁡Asin⁡B\sin A\sin B identity?
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It is sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A\sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)], with a minus sign; the cosine product has a plus. Mixing them flips a sign through the whole answer.

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