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Intro to Analysis in-class_Part_42

# Intro to Analysis in-class_Part_42 - 6.2 Properties of the...

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6.2 Properties of the Riemann Integral 89 Putting this minor result together with the previous two shows that D ( I ( f )) = f, but I ( D ( f )) = f + c, so integration and diﬀerentiation are almost inverse operations. Theorem 6.2.6 (Integration by parts) . If f,g C 1 [ a,b ] , then Z b a f ( x ) g 0 ( x ) dx = f ( b ) g ( b ) - f ( a ) g ( a ) - Z b a f 0 ( x ) g ( x ) dx. Proof. Put h ( x ) = f ( x ) g ( x ), so h,h 0 ∈ R by Integral properties thm. Use the Integration of derivative thm on h 0 . NOTE: IBP is product rule in reverse, just like CoV is chain rule in reverse. Theorem 6.2.7 (Change of variable) . If g C 1 [ a,b ] , g is increasing, and f ∈ R [ g ( a ) ,g ( b )] , then f g ∈ R [ a,b ] and Z b a f ( g ( x )) g 0 ( x ) dx = Z g ( b ) g ( a ) f ( y ) dy. Proof. First, show f g ∈ R . To each partition P = { x 0 ,...,x n } of [ a,b ], there corresponds a partition Q = { y 0 ,...,y n } of [ g ( a ) ,g ( b )], so that g ( x i ) = y i . Since the range of f on [ y i - 1 ,y

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Intro to Analysis in-class_Part_42 - 6.2 Properties of the...

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