Chap7_Sec3 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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Unformatted text preview: 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION 7.3 Trigonometric Substitution In this section, we will learn about: The various types of trigonometric substitutions. 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 TRIGONOMETRIC SUBSTITUTION If we change the variable from x to by the substitution x = a sin , the identity 1 sin 2 = cos 2 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 - =- =- = = TRIGONOMETRIC SUBSTITUTION Notice the difference between the substitution u = a 2 x 2 and the substitution x = a sin . 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. 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. 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 = INVERSE SUBSTITUTION We can make the inverse substitution x = a sin , provided that it defines a one-to-one function. This can be accomplished by restricting to lie in the interval [- /2, /2]. TABLE OF TRIGONOMETRIC SUBSTITUTIONS Here, we list trigonometric substitutions that are effective for the given radical expressions because of the specified trigonometric identities. TABLE OF TRIGONOMETRIC SUBSTITUTIONS In each case, the restriction on 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. TRIGONOMETRIC SUBSTITUTION Evaluate Let x = 3 sin , where /2 /2. Then, dx = 3 cos d and Note that cos 0 because /2 /2.) Example 1 2 2 9- x dx x 2 2 2 9 9 9sin 9cos 3 cos 3cos - =- = = = x 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 - = = = =- = -- + x dx d x d d d C Example 1 TRIGONOMETRIC SUBSTITUTION...
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This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

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Chap7_Sec3 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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