int_by_parts - Harvey Mudd College Math Tutorial:...

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Harvey Mudd College Math Tutorial: Integration by Parts We will use the Product Rule for derivatives to derive a powerful integration formula: Start with ( f ( x ) g ( x )) 0 = f ( x ) g 0 ( x ) + f 0 ( x ) g ( x ). Integrate both sides to get f ( x ) g ( x ) = Z f ( x ) g 0 ( x ) dx + Z f 0 ( x ) g ( x ) dx . (We need not include a constant of integration on the left, since the integrals on the right will also have integration constants.) Solve for Z f ( x ) g 0 ( x ) dx , obtaining Z f ( x ) g 0 ( x ) dx = f ( x ) g ( x ) - Z f 0 ( x ) g ( x ) dx. This formula frequently allows us to compute a difficult integral by computing a much simpler integral. We often express the Integration by Parts formula as follows: Let u = f ( x ) dv = g 0 ( x ) dx du = f 0 ( x ) dx v = g ( x ) Then the formula becomes Z udv = uv - Z v du. To integrate by parts, strategically choose u , dv and then apply the formula. Example
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This note was uploaded on 04/06/2011 for the course ECON 3190 taught by Professor Hong during the Fall '07 term at Cornell University (Engineering School).

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int_by_parts - Harvey Mudd College Math Tutorial:...

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