Lesson 12Integration By Trigonometric Substitution

Lesson 12Integration By Trigonometric Substitution - , such...

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Click to edit Master subtitle style TOPIC TECHNIQUES OF INTEGRATION
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Click to edit Master subtitle style TECHNIQUES OF INTEGRATION 1. Integration by parts 2. Integration by trigonometric substitution 3. Integration by miscellaneous substitution 4. Integration by partial fraction
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Click to edit Master subtitle style TECHNIQUES OF INTEGRATION 2. Integration by trigonometric substitution
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OBJECTIVES To evaluate integrals using trigonometric substitution
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Integration by Trigonometric Substitution : If the integrand contains integral powers of x and an expression of the form , , and where a > 0, it is often possible to perform the integration by using a trigonometric substitution which results to an integral involving trigonometric functions. 2 2 u a - 2 2 u a + 2 2 a u - There are three cases considered depending on the radical contained in the integrand. Each form shall be discussed separately.
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Case 1: Integrand containing radical of the form where a > 0. 2 2 u a - u E a 2 2 u a -
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Introduce a new variable
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Unformatted text preview: , such that: Obj106 Obj107 = sin a u =-cos a u a 2 2 = d cos a du EXAMPLE: -2 2 x 4 9 x dx . 1 Evaluate the given integral. -4 2 3 x 16 dx x . 2 Case 2: Integrand containing radical of the form where a > 0. 2 2 u a + u E 2 2 u a + a a u tan = = tan a u = d sec a du 2 Le t a u a sec 2 2 + = = + sec a u a 2 2 EXAMPLE ( 29 + 2 / 3 2 x 3 4 dx . 1 + 2 y 4 25 y dy . 2 Case 3: Integrand containing radical of the form where a > 0. 2 2 a u-u E 2 2 a u-a Introduce a new variable , such that: a u sec = Le t = sec a u = d tan sec a du a a u tan 2 2-= =-tan a a u 2 2 Classwork: ( 29 ( 29 -4 x ln x dx x ln . 1 2 3 ( 29 + + 2 / 3 x x 2 x 7 e 8 e dx e . 2 HOMEWORK 2-3: Evaluate the following -2 2 x 8 4 x dx . 1 -dx x 2 x 9 16 . 2 2 ( 29 -2 / 3 2 x 4 5 dx . 3 ( 29 + 2 2 w 25 dw . 4 ( 29 -2 / 3 2 x x 6 dx . 5...
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This note was uploaded on 08/19/2011 for the course ECON 232 taught by Professor Charles during the Spring '11 term at MIT.

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Lesson 12Integration By Trigonometric Substitution - , such...

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