n8b - Section 8.3 Trigonometric Substitution courtesy:...

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Unformatted text preview: Section 8.3 Trigonometric Substitution courtesy: Chia-Rong Chen 1. When the integrand involves a 2- x 2 , a 2 + x 2 , or x 2- a 2 , and the integral cannot be evaluated by substitution method, we make use of the identities: 1- sin 2 = cos 2 , 1 + tan 2 = sec 2 , sec 2 - 1 = tan 2 . 2. trigonometric substitutions: expression substitution range new expression final substitution a 2- x 2 x = a sin - 2 2 a 2 cos 2 = sin- 1 x a a 2 + x 2 x = a tan - 2 < < 2 a 2 sec 2 = tan- 1 x a x 2- a 2 x = a sec < 2 , a 2 tan 2 = sec- 1 x a or < 3 2 eg.(1) Evaluate integraldisplay dx x 2 x 2- 16 dx . 1 eg.(2) integraldisplay dx a 2 + x 2 eg.(3) integraldisplay x 2 9- x 2 dx 2 eg.(4) integraldisplay 1 x 3 x 2 + 1 dx eg.(5) integraldisplay dx x 2 + 4 x + 8 3 . eg.(6) integraldisplay e 2 t- 9 dt 4 eg.(7) integraldisplay a 2- x 2 dx 5 Section 8.4 Integration of Rational Functions by Partial Fractions...
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This note was uploaded on 02/11/2009 for the course ENTC 219 taught by Professor Staff during the Spring '08 term at Texas A&M.

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n8b - Section 8.3 Trigonometric Substitution courtesy:...

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