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Integrals Involving Powers of Sine and Cosine

Integrals Involving Powers of Sine and Cosine. Integrals Involving Powers of Sine and Cosine. In this section you will study techniques for evaluating integrals of the form where either m or n is a positive integer.

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Integrals Involving Powers of Sine and Cosine

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  1. Integrals Involving Powers of Sine and Cosine

  2. Integrals Involving Powers of Sine and Cosine In this section you will study techniques for evaluating integrals of the form where either m or n is a positive integer. To find antiderivatives for these forms, try to break them into combinations of trigonometric integrals to which you can apply the Power Rule.

  3. These will be used to integrate powers of sine and cosine Basic Identities • Pythagorean Identities • Half-Angle Formulas

  4. Integrals Involving Powers of Sine and Cosine

  5. Try with Let u=sinx du=cosx dx

  6. Example – Power of Sine Is Odd and Positive Find Solution: Because you expect to use the Power Rule with u = cos x, save one sine factor to form du and convert the remaining sine factors to cosines.

  7. Example – Solution cont’d

  8. Integrals Involving Powers of Sine and Cosine For instance, you can evaluate  sin5 xcosx dx with the Power Rule by letting u = sin x. Then, du = cosxdx and you have To break up  sinmxcosnx dx into forms to which you can apply the Power Rule, use the following identities.

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

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

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

  12. 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 • If the powers of both sine and cosine are • even, use the power reducing formulas:

  13. Combinations of sin, cos • Consider • Use Pythagorean identity

  14. Combinations of sin, cos u=cos4x du= -4sin4x dx

  15. Integrals Involving Powers of Secant and Tangent

  16. Integrals Involving Powers of Secant and Tangent The following guidelines can help you evaluate integrals of the form

  17. Integrals Involving Powers of Secant and Tangent cont’d

  18. Combinations of tanm, secn • Try factoring out sec2 x or tan x sec x

  19. Integrals of Even Powers of sec, csc • Use the identity sec2 x – 1 = tan2 x • Try u=tan3x du=3sec^2(3x) dx

  20. Homework • Section 8.3, pg. 540: 5-17 odd, 25-29 odd

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