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Trigonometric Integrals

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

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Trigonometric Integrals

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  1. Trigonometric Integrals Lesson 8.3

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

  3. 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

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

  5. Integral of sinn x, nEven • Use half-angle formulas • Try Change to power of cos2 x • Expand the binomial, then integrate

  6. Try with 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

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

  8. 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

  9. 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 ofcotm x and cscn x

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

  11. 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 cosnx is replaced by sinnx

  12. Wallis's Formulas • Try it out …

  13. Assignment • Lesson 8.3 • Page 540 • Exercises 1 – 41 EOO

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