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Chap7_Sec3 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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7 TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION
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TECHNIQUES OF INTEGRATION 7.3 Trigonometric Substitution In this section, we will learn about: The various types of trigonometric substitutions.
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TRIGONOMETRIC SUBSTITUTION In finding the area of a circle or an ellipse, an integral of the form arises, where a > 0. If it were , the substitution would be effective. However, as it stands, is more difficult. 2 2 a x dx - 2 2 - x a x dx 2 2 = - u a x 2 2 - a x dx
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TRIGONOMETRIC SUBSTITUTION If we change the variable from x to θ by the substitution x = a sin θ , the identity 1 – sin 2 θ = cos 2 θ lets us lose the root sign. This is because: 2 2 2 2 2 2 2 2 2 sin (1 sin ) cos cos a x a a a a a θ θ θ θ - = - = - = =
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TRIGONOMETRIC SUBSTITUTION Notice the difference between the substitution u = a 2 x 2 and the substitution x = a sin θ . In the first, the new variable is a function of the old one. In the second, the old variable is a function of the new one.
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TRIGONOMETRIC SUBSTITUTION In general, we can make a substitution of the form x = g ( t ) by using the Substitution Rule in reverse. To make our calculations simpler, we assume g has an inverse function, that is, g is one-to-one.
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INVERSE SUBSTITUTION Here, if we replace u by x and x by t in the Substitution Rule (Equation 4 in Section 5.5), we obtain: This kind of substitution is called inverse substitution. ( ) ( ( )) '( ) f x dx f g t g t dt =
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INVERSE SUBSTITUTION We can make the inverse substitution x = a sin θ , provided that it defines a one-to-one function. This can be accomplished by restricting θ to lie in the interval [- π /2, π /2].
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TABLE OF TRIGONOMETRIC SUBSTITUTIONS Here, we list trigonometric substitutions that are effective for the given radical expressions because of the specified trigonometric identities.
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TABLE OF TRIGONOMETRIC SUBSTITUTIONS In each case, the restriction on θ is imposed to ensure that the function that defines the substitution is one-to-one. These are the same intervals used in Section 1.6 in defining the inverse functions.
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TRIGONOMETRIC SUBSTITUTION Evaluate Let x = 3 sin θ , where – π /2 ≤ θ π /2. Then, dx = 3 cos θ and Note that cos θ ≥ 0 because – π /2 ≤ θ π /2.) Example 1 2 2 9 - x dx x 2 2 2 9 9 9sin 9cos 3 cos 3cos θ θ θ θ - = - = = = x
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TRIGONOMETRIC SUBSTITUTION Thus, the Inverse Substitution Rule gives: 2 2 2 2 2 2 2 9 3cos 3cos 9sin cos sin cot (csc 1) cot θ θ θ θ θ θ θ θ θ θ θ θ θ - = = = = - = - - + x dx d x d d d C Example 1
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TRIGONOMETRIC SUBSTITUTION As this is an indefinite integral, we must return to the original variable x.
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