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TRIGONOMETRIC SUBSTITUTION Find : cos , sin , tan 3 9 x2 x TRIGONOMETRIC SUBSTITUTION 4 x x 2 sin Let : 2 4 x 2 4 (2 sin ) 2 4 4 sin 2 4(1 sin ) NOTE: We convert the radical into a cos function 2 4 cos 2 2 cos 2 cos if : π 2 θ π 2 TRIGONOMETRIC SUBSTITUTION x 25 Let : x 5 tan x 9 Let : x 3sec 2 2 TRIGONOMETRIC SUBSTITUTION Example Example Find Example 4 x 2 dx Find 2 x 4 x dx Find 9 x2 dx 2 x 1 dx x 2 16 x 1 2 4 x2 1 x 2 16 dx dx The main idea is how to get rid of the radicals. ( use trig. Substitution) TRIGONOMETRIC SUBSTITUTION Expression a2 x2 a x 2 2 x2 a2 x a sin x a sin1 sin 2 cos 2 x a tan x a tan x a tan x a sec x a sec x a sec TRIGONOMETRIC SUBSTITUTION Example Find 1 9 x 2 dx In the following table we list trigonometric substitutions that are effective for the given radical expressions because of the specified trigonometric identities. TRIGONOMETRIC SUBSTITUTION Example Find 3 2 x 9 x dx In the following table we list trigonometric substitutions that are effective for the given radical expressions because of the specified trigonometric identities. TRIGONOMETRIC SUBSTITUTION Example Find 9 x2 dx 2 x TRIGONOMETRIC SUBSTITUTION Example Find x 3dx (4 x 2 9)3/ 2 x 32 tan TRIGONOMETRIC SUBSTITUTION EXAM-2 Term-092 TRIGONOMETRIC SUBSTITUTION Example Find x x 4 2 dx TRIGONOMETRIC SUBSTITUTION EXAM-2 Term-102 TRIGONOMETRIC SUBSTITUTION Example Find dx x2 a2 a0 x a cosh t hyperbolic substitutions can be used in place of trigonometric substitutions and sometimes they lead to simpler answers. But we usually use trigonometric substitutions because trigonometric identities are more familiar than hyperbolic identities. TRIGONOMETRIC SUBSTITUTION Example Find x 3 2x x 2 dx