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SEC 8.2: TRIGONOMETRIC INTEGRALS Example Find cos x dx 3 3 2 cos x dx cos x cos xdx (1 sin 2 x) cos xdx Example Find 5 2 sin x cos x dx 5 2 4 2 sin x cos x dx sin x cos x sin x dx (1 cos 2 x) 2 cos 2 x sin xdx TRIGONOMETRIC INTEGRALS sin m n x cos x dx sin odd m is odd cos odd n is odd save one sin 1 save one cos with dx 1 2 use cos x 1- sin x 2 use sin x 1- cos x 2 2 to express the remaining factors in terms of sin 3 2 cos x dx cos x cos xdx 2 2 to express the remaining factors in terms of cos 5 2 sin x cos x dx sin 4 x cos 2 x sin x dx TRIGONOMETRIC INTEGRALS odd cos even even odd odd odd even even sin sin even cos even 4 sin dx 1 4 2 1 cos 2 x dx 1 use half angle sin 2 x 12 (1- cos 2 x) cos2 x 12 (1 cos 2 x) 2 sometimes helpful to use sin x cos x 12 sin 2 x TRIGONOMETRIC INTEGRALS Eliminating Square Roots we use the identity cos2 x 12 (1 cos 2 x) to eliminate a square root. Example Find 4 0 1 cos 4 x dx TRIGONOMETRIC INTEGRALS We can use a similar strategy to evaluate integrals of the form tan m n x sec x dx Example Find tan 6 4 x sec x dx u tan x du sec 2 xdx sec 2 x 1 tan 2 x Example Find 5 4 tan x sec x dx u sec x du sec x tan xdx tan 2 x 1 sec 2 x TRIGONOMETRIC INTEGRALS tan m n x sec x dx tan odd m is odd sec even n is even 1 save one sec 2 2 use sec x 1 tan x 2 to express the remaining factors in terms of tan 2 1 save one sec x tan x 2 use tan 2 x sec 2 x 1 to express the remaining factors in terms of sec EXAM-2 Term-082 TRIGONOMETRIC INTEGRALS tan even tan sec odd even even odd odd odd even even sec odd the guidelines are not as clear-cut. We may need to use identities, integration by parts, and occasionally a little ingenuity. TRIGONOMETRIC INTEGRALS tan even sec odd Example Find 3 sec xdx the guidelines are not as clear-cut. We may need to use identities, integration by parts, and occasionally a little ingenuity. If an even power of tangent appears with an odd power of secant, it is helpful to express the integrand completely in terms of sec x Powers of sec x may require integration by parts, as shown in the following example. TRIGONOMETRIC INTEGRALS Example Find 3 sec xdx TRIGONOMETRIC INTEGRALS REMARK Integrals of the form m n cot x csc x dx can be found by similar methods because of the identity 1 cot 2 x csc 2 x cot x csc x dx m n cot odd m is odd csc even n is even 1 save one sec 2 2 csc x 1 cot x 2 2 to express the remaining factors in terms of cot 1 save one csc x cot x 2 use cot 2 x csc 2 x 1 to express the remaining factors in terms of csc EXAM-2 Term-122 Product of Sines and Cosines cos mx cos nx dx sin mx sin nx dx sin mx cos nx dx EXAM-2 Term-122 TRIGONOMETRIC INTEGRALS Powers of Sines and Cosines Products of Sines and Cosines TRIGONOMETRIC INTEGRALS Powers of tan x and sec x Eliminating Square Roots EXAM-2 Term-092 EXAM-2 Term-092 TRIGONOMETRIC INTEGRALS function of tan and sec function of Sines and Cosines xdx f (cos x)sin 2 f (tan x)sec xdx du du x tan xdx f (sec x)sec xdx f (sin x)cos du du TRIGONOMETRIC INTEGRALS function of cot and csc 2 f (cot x)csc xdx du x cot xdx f (csc x)csc du EXAM-2 Term-092 EXAM-2 Term-092 EXAM-2 Term-092 EXAM-2 Term-092