1120_2 - Week 2 Outline Integration by parts The formula...

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Week 2 Outline Integration by parts – The formula – How to decide what should be u and what should be d v – Reduction formula The Inverse Trig Substitution – Integrals involving a 2 - x 2 – Integrals involving a 2 + x 2 or 1 x 2 + a 2 – Integrals involving x 2 - a 2 Completing the square Integrals involving n ax + b Multiplying by a form of 1 The tan θ 2 substitution Integration by parts If we reverse the product rule ( u ( x ) v ( x )) 0 = u 0 ( x ) v ( x ) + u ( x ) v 0 ( x ) to do integration, we will get the formula of integration by parts Z u ( x ) v 0 ( x ) d x = u ( x ) v ( x ) - Z u 0 ( x ) v ( x ) d x. By using the shortened forms d u = d u d x d x = u 0 d x and d v = d v d x d x = v 0 d x, we get the usual form for integration by parts Z u d v = uv - Z v d u. Remark The trick in applying this powerful method is to decide which part of the original integral should be u and which part should be d v . 1
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Examples 1. R x ln x d x 2. R ln x d x 3. R tan - 1 x d x 4. R x 2 tan - 1 x d x 5. R x 2 e x d x 6. R e x sin x d x How to decide what should be u and what should be d v Rule 1. The d v has to be something one can integrate. The u is what’s left over.
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