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Integration Techniques, L’Hôpital’s Rule, and Improper Integrals 8 Copyright © Cengage Learning. All rights reserved.

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2 Trigonometric Integrals Not tested, 1 day only Copyright © Cengage Learning. All rights reserved. 8.3

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3 Integrals Involving Powers of Sine and Cosine

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4 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.

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5 Integrals Involving Powers of Sine and Cosine For instance, you can evaluate sin 5 x cos x dx with the Power Rule by letting u = sin x. Then, du = cos x dx and you have To break up sin m x cos n x dx into forms to which you can apply the Power Rule, use the following identities.

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

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7 Integrals Involving Powers of Sine and Cosine

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8 Integral of sin n x, n Odd Split into product of an even power 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

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

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

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

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12 Try with Let u=sinx du=cosx dx

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13 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.

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14 Example – Solution cont’d

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15 Combinations of sin, cos Consider Use Pythagorean identity

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

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17 Integrals Involving Powers of Secant and Tangent

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

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19 Integrals Involving Powers of Secant and Tangent cont’d

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20 Example 4 – Power of Tangent Is Odd and Positive Find Solution: Because you expect to use the Power Rule with u = sec x, save a factor of (sec x tan x) to form du and convert the remaining tangent factors to secants.

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21 Example 4 – Solution cont’d

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22 Combinations of tan m, sec n Try factoring out sec 2 x or tan x sec x

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23 Integrals of Even Powers of sec, csc Use the identity sec 2 x – 1 = tan 2 x Try u=tan3x du=3sec^2(3x) dx

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

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25 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

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26 Wallis's Formulas Try it out …

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27 These formulas are also valid if cos n x is replaced by sin n x. Integrals Involving Powers of Sine and Cosine

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28 Integrals Involving Sine-Cosine Products with Different Angles

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29 Integrals Involving Sine-Cosine Products with Different Angles Integrals involving the products of sines and cosines of two different angles occur in many applications. In such instances you can use the following product-to-sum identities.

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30 Example 8 – Using Product-to-Sum Identities Find Solution: Considering the second product-to-sum identity above, you can write

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