Packet6TechniquesOFIntegrationByPartsANDTrigIntegrals.docx - Packet 6 \u2013 Calculus II Techniques of Integration Integration by Parts the shoe-string

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Packet 6 – Calculus II Techniques of Integration - Integration by Parts (& the shoe-string method ) Recall the Product Rule for differentiation: d dx [ f ( x ) ∙ g ( x ) ] = f ' ( x ) ∙ g ( x ) + g ' ( x ) ∙f ( x ) Rewrite using integration notation: Rearrange ( Solve for f ( x ) ∙g' ( x ) dx ): Integration by Parts Indefinite Integrals: udv = uv v du Definite Integrals: a b udv = uv | a b a b v du Examples: Indefinite Integrals a) x cos ( 5 x ) dx
(now show using shoe string method) b) ln x dx
(now show using shoe string method) c) y e 0.2 y dy
(now show using shoe string method) d) t sinh ( mt ) dt
(now show using shoe string method) Example: Definite Integrals a) 0 1 ( x 2 + 1 ) e x dx
(now show using shoe string method) b) 1 3 arctan ( 1 x ) dx (now show using shoe string method)
- Trigonometric Integrals a) sin 3 x dx b) sin 3 x cos 4 x dx
c) tan 2 θ sec 4 θdθ Example: