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Math 1432 Bubble in PS ID and Popper Number very carefully. If you make a bubbling mistake, your scantron can’t be saved in the system. In that case, you will not get credit for the popper even if you turned it in. Check your CASA account for Quiz due dates. Don’t miss any online quizzes! Be considerate of others in class. Respect your friends and do not distract anyone during the lecture Section 8.2 Powers and Products of Trigonometric Functions Recall the following identities: cos 2 x sin 2 x 1 1 tan 2 x sec 2 x 1 cot 2 x csc 2 x cos 2 x 1 cos 2x 2 sin 2 x 1 cos 2x 2 sin 2x 2 sin x cos x cos 2x cos 2 x sin 2 x 1 In this section, we will study techniques for evaluating integrals of the form sin m x cos x dx n and sec m x tan n dx where either m or n is a positive integer. To find antiderivatives for these forms, try to break them into combinations of trigonometric integrals to which you can apply a formula: u n1 C u n du n 1 ln u C if n 1 if n 1 . Integrals Involving Powers of Sine and Cosine 1. If m or n odd: 2 a. m odd: rewrite identity sin m x as sin m-1 xsin x (m-1 is even so can use sin 2 x 1 cos 2 x ) cos m-1 x cos x (n-1 is even so can use cos 2 x 1 sin 2 x ) m b. n odd: rewrite cos x as identity Example: sin 3 xdx Example: sin 3 x cos 2 xdx 3 Example: cos 5 xdx Example: sin 4 x cos 5 xdx 4 2. If m and n even use these identities: sin 2 x 1 cos 2x 2 and /or cos 2 x 1 cos 2x 2 . Example: cos 2 xdx 5 Integrals involving Secants and Tangents tan 2 x 1 sec 2 x For m n tan xsec xdx ò a. n even: rewrite can use identity tan m xsecn x as tan m xsecn-2 xsec2 x (then you sec 2 x tan 2 x 1 ) tanm-1 xsecn-1 x ×sec x tan x (m-1 is even so 2 2 can use identity tan x = sec x -1) 2 2 m m even and n odd: rewrite tan x using tan x = sec x -1 b. m odd: rewrite as c. Example: tan 3 xdx 6 Example: sec 4 xdx Example: sec 4 x tan 2 xdx 7 Example: sec x tan 2 x dx POPPER# 8 Q# ò cos x sin 3 x dx a. 1 cos2 x + C 2 b. c. 1 4 sin x + C 4 d. none of these 1 cos 4 x + C 4 9