Trigonometric Integrals Lesson 8.3

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

Integral of sin n x, n Odd Split into product of an even and sin x Make the even power a power of sin 2 x Use the Pythagorean identity Let u = cos x, du = -sin x dx

Integral of sin n x, n Odd Integrate and un-substitute Similar strategy with cos n x, n odd

Integral of sin n x, n Even Use half-angle formulas Try Change to power of cos 2 x Expand the binomial, then integrate

Combinations of sin, cos 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 Try with

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

Combinations of tan m, sec n When n is even  Factor out sec 2 x  Rewrite remainder of integrand in terms of Pythagorean identity sec 2 x = 1 + tan 2 x  Then u = tan x, du = sec 2 x dx Try

Combinations of tan m, sec n When m is odd  Factor out tan x sec x (for the du)  Use identity sec 2 x – 1 = tan 2 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 cot m x and csc n x

Integrals of Even Powers of sec, csc Use the identity sec 2 x – 1 = tan 2 x Try

Wallis's Formulas If n is odd and (n ≥ 3) then If n is even and (n ≥ 2) then And … Believe it or not These formulas are also valid if cos n x is replaced by sin n x And … Believe it or not These formulas are also valid if cos n x is replaced by sin n x

Wallis's Formulas Try it out …

Assignment Lesson 8.3 Page 540 Exercises 1 – 41 EOO