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
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
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 -
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 &gt; 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 &gt; 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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