Lesson 11 - Integration By Trigonometric Substitution

Lesson 11 - Integration By Trigonometric Substitution - =...

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TOPIC TECHNIQUES OF INTEGRATION
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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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TECHNIQUES OF INTEGRATION 2. Integration by trigonometric substitution
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OBJECTIVES 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 θ a 2 2 u a -
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Introduce a new variable , such that: θ a u sin = θ a u a cos 2 2 - = θ θ
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Unformatted text preview: = 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 2 2 u a + a a u tan = = tan a u = d sec a du 2 Let 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 2 2 a u-a Introduce a new variable , such that: a u sec = Let = sec a u = d tan sec a du a a u tan 2 2-= =-tan a a u 2 2 EXAMPLE ( 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 -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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Lesson 11 - Integration By Trigonometric Substitution - =...

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