More Trigonometric Integrals Lesson 9.4

Recall Basic Identities Pythagorean Identities Half-Angle Formulas These will be used to integrate powers of sin and cos

Integral of sinn x, n Odd Split into product of an even and sin x Make the even power a power of sin2 x Use the Pythagorean identity Let u = cos x, du = -sin x dx

Integral of sinn x, n Odd Integrate and un-substitute Similar strategy with cosn x, n odd

Integral of sinn x, n Even Use half-angle formulas Try Change to power of cos2 x Expand the binomial, then integrate

Combinations of sin, cos Try with General form 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

Combinations of sin, cos Consider Use Pythagorean identity Separate and use sinn x strategy for n odd

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

Combinations of tanm, secn When m is odd Factor out tan x sec x (for the du) Use identity sec2 x – 1 = tan2 x for even powers of tan x Let u = sec x, du = sec x tan x Try the same integral with this strategy Note similar strategies for integrals involving combinations of cotm x and cscn x

Integrals of Even Powers of sec, csc Use the identity sec2 x – 1 = tan2 x Try

Assignment Lesson 9.4 Page 376 Exercises1 – 33 odd