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6.2 Trigonometric Integrals How to integrate powers of sinx and cosx (i) If the power of cos x is odd, save one cosine factor and use cos2x = 1 - sin2x to express the remaining factors in terms of sin x. Then substitute u = sin x. (ii) If the power of sin x is odd, save one sine factor and use sin2x = 1 - cos2x to express the remaining factors in terms of cos x. Then substitute u = cos x. (iii) If the powers of both sine and cosine are even, use the half-angle identities: sin2x = 0.5(1 – cos 2x) cos2x = 0.5(1 + cos 2x) It is sometimes helpful to use the identity: sin x cos x = 0.5 sin 2x Example: Evaluate the integral (the solution on the board) sin 6 3 x cos x dx How to integrate powers of tanx and secx (i) If the power of sec x is even, save a factor of sec2x and use sec2x = 1 + tan2x to express the remaining factors in terms of tan x. Then substitute u = tan x. (ii) If the power of tan x is odd, save a factor of sec x tan x and use tan2x = sec2x – 1 to express the remaining factors in terms of sec x. Then substitute u = sec x. Example: Evaluate the integral (the solution on the board) 3 tan x sec x dx