n8a - Section 8.1 Integration by Parts courtesy: Chia-Rong...

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Unformatted text preview: Section 8.1 Integration by Parts courtesy: Chia-Rong Chen 1. integration-by-parts formula : Start with the product rule, d dx ( u v ) = u dv dx + v du dx and rewrite it in the form, d ( u v ) = u dv + v du Thus, u dv = d ( u v )- v du. We then integrate both sides to get integraldisplay u dv = u v- integraldisplay v du integraldisplay b a u dv = [ u v ] b a- integraldisplay b a v du 2. rules: For u , choose the one whose derivative is simpler. For v , choose the one whose integral is simpler. eg.(1) integraldisplay ln x dx eg.(2) integraldisplay x cos x dx eg.(3) integraldisplay 2 1 x 3 ln x dx 1 eg.(4) integraldisplay x 2 e- x dx eg.(5) integraldisplay e x cos x dx 2 eg.(6) integraldisplay 2 1 x sec- 1 x dx 3 Section 8.2 Trigonometric Integrals courtesy: Chia-Rong Chen 1. integraldisplay sin m x cos n x dx , odd powers: (a) m = 2 k +1: Factor out one sine and use sin 2 x = 1- cos 2 x . Then substitute u = cos x ....
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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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n8a - Section 8.1 Integration by Parts courtesy: Chia-Rong...

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