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Unformatted text preview: Lecture 1: MATH 138-W12-003 January 6, 2012 I) Integration by Parts, Section 7.1 ”I never met an integral I couldn’t integrate by parts” Recall that the product rule states, d dx [ f ( x ) g ( x )] = f ( x ) g ( x ) + f ( x ) g ( x ) . If we rearrange the terms by isolating the first term on the right and then integrate the equation we get a new and very useful technique for integrating functions, f ( x ) g ( x ) = d dx [ f ( x ) g ( x )]- f ( x ) g ( x ) , Z f ( x ) g ( x ) dx = Z d dx [ f ( x ) g ( x )] dx- Z f ( x ) g ( x ) dx, Z f ( x ) g ( x ) dx = f ( x ) g ( x )- Z f ( x ) g ( x ) dx. Therefore we have proven that, Z f ( x ) g ( x ) dx = f ( x ) g ( x )- Z f ( x ) g ( x ) dx . This last equation above is known and the formula for integration by parts . If we set u = f ( x ) and v = g ( x ) then it can be more compactly written as (using substitution), Z udv = uv- Z vdu . If we have bounds on the integral we do the same thing but now we have to evaluate the first term on the right at the two bounds,...
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