Lesson 8 InverseTrigFunctions-Integration

Lesson 8 - the arctan function 2 2 10 dx x x Completing the Square • Try these 2 2 2 4 13 dx x x 2 2 4 dx x x Example Evaluate the

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INTEGRATION OF INVERSE TRIGONOMETRIC FUNCTIONS
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Review Recall derivatives of inverse trig functions 2 1 2 1 2 1 2 1 sin , 1 1 1 tan 1 1 sec , 1 1 d du u u dx dx u d du u dx u dx d du u u dx dx u u - - - = < - = + = < -
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Integrals Using Same Relationships 3 2 2 2 2 2 2 arcsin 1 arctan 1 arcsec du u C a a u du u C a u a a du u C a a u u a = + - = + + = + - When given integral problems, look for these patterns
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. If a=1, you have: 1. + = - - C u sin u 1 du 1 2 2. + = + - C u tan u 1 du 1 2 3. + = - - C u u u du 1 2 sec 1
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Identifying Patterns For each of the integrals below, which inverse trig function is involved? 5 2 4 13 16 dx x + 2 25 4 dx x x - 2 9 dx x -
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Warning Many integrals look like the inverse trig forms Which of the following are of the inverse trig forms? 6 2 1 dx x + 2 1 x dx x + 2 1 dx x - 2 1 x dx x - If they are not, how are they integrated?
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Try These Look for the pattern or how the expression can be manipulated into one of the patterns 7 2 8 1 16 dx x + 2 1 25 x dx x - 2 4 4 15 dx x x - + + 2 5 10 16 x dx x x - - +
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Completing the Square Often a good strategy when quadratic functions are involved in the integration Remember … we seek (x – b) 2 + c Which might give us an integral resulting in
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Unformatted text preview: the arctan function 2 2 10 dx x x + + ∫ Completing the Square • Try these 2 2 2 4 13 dx x x-+ + ∫ 2 2 4 dx x x-+ ∫ Example : Evaluate the following integrals using the formulas for integrals yielding inverse trigonometric functions: ∫-2 5 2 . 1 s ds ∫-2 2 . 2 x x dx ∫--2 3 . 3 2 x x dx ∫-9 16 . 4 2 y y dy ∫ + + 1 . 5 2 x x xdx ( 29 ∫ + + + 3 4 2 3 . 6 2 x x x dx ( 29 ∫ + +-3 2 2 1 3 . 7 2 x x dx x HOMEWORK : Evaluate the following integral . 11 ∫-4 9 16 . 1 r rdr ∫-x xdx 2 cos 2 sin . 2 ( 29 ∫ + x x dx 1 . 3 ∫ + + dx x x 2 2 4 1 . 4 ∫-dx e e x x 2 1 . 5 ∫--2 2 3 . 6 x x xdx θ d ∫ + 2 sin 1 cos . 7 ∫ +-3 4 2 2 . 8 2 3 x x dx x ( 29 [ ] ∫ + 2 ln 1 . 9 x x dx ∫-1 9 . 10 2 x x dx ∫-1 2 2 . 11 t dt ∫-+ 1 2 / 1 2 4 4 3 . 12 dx x x ∫--3 ln 2 ln . 13 z z e e dz...
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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 8 - the arctan function 2 2 10 dx x x Completing the Square • Try these 2 2 2 4 13 dx x x 2 2 4 dx x x Example Evaluate the

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