More Trigonometric Integrals Lesson 9.4 Recall Basic Identities • Pythagorean Identities sin 2 cos 2 1 tan 2 1 sec 2 1 cot csc 2 2 • Half-Angle Formulas 1 cos 2 sin 2 1 cos 2 2 cos 2 These will be used to integrate powers of sin and cos 2 2 Integral of sinn x, n Odd • Split into product of an even and sin x 5 4 sin x dx sin x sin x dx • Make the even power a power of sin2 x sin x sin x dx sin x sin x dx 4 2 2 • Use the Pythagorean identity sin x 2 2 sin x dx 1 cos x sin x dx 2 2 • Let u = cos x, du = -sin x dx 1 u 2 2 du 1 2u u du ... 2 4 3 Integral of sinn x, n Odd • Integrate and un-substitute 2 3 1 5 1 2u u du u u u C 3 5 2 1 3 cos x cos x cos5 C 3 5 2 4 • Similar strategy with cosn x, n odd 4 Integral of sinn x, n Even • Use half-angle formulas 1 cos 2 sin 2 2 4 cos 5x dx Change to power of cos2 x • Try cos 2 2 2 1 dx 1 cos10 x dx 2 • Expand the binomial, then integrate 5 Combinations of sin, cos • General form Try with n x dx sinsin x x cos cos x dx m 2 3 • If either n or m is odd, use techniques as before Split the odd power into an even power and power of one Use Pythagorean identity Specify u and du, substitute Usually reduces to a polynomial Integrate, un-substitute 6 Combinations of sin, cos • Consider sin 3 4 x cos x dx 2 • Use Pythagorean identity sin 3 4 x 1 sin 4 x dx sin 4 x sin 4 x dx 2 3 5 • Separate and use sinn x strategy for n odd 7 Combinations of tanm, secn • When n is even Factor out sec2 x Rewrite remainder of integrand in terms of Pythagorean identity sec2 x = 1 + tan2 x Then u = tan x, du = sec2x dx • Try sec y tan y dy 4 3 8 Combinations of tanm, secn • When m is odd Note similar Factor out tan x sec x (for the du)strategies for integrals involving Use identity sec2 x – 1 = tan2combinations x for evenof cotm x and cscn x powers of tan x Let u = sec x, du = sec x tan x • Try the same integral with this strategy sec 4 y tan y dy 3 9 Integrals of Even Powers of sec, csc • Use the identity sec2 x – 1 = tan2 x • Try sec 4 3x dx sec 3x sec 3x dx 1 tan 3x sec 3x dx sec 3x tan 3x sec 3x dx 2 2 2 2 2 2 2 1 3 1 tan 3 x tan 3 x C 9 3 10 Assignment • Lesson 9.4 • Page 376 • Exercises1 – 33 odd 11