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Lecture

7.1 Integration by Parts: Overview

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Department
Mathematics
Course Code
MAT136H1
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all

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7.1 Integration Techniques
Integration by Parts: Overview
Indefinite Integral
This follows after the form of the Product Rule:   
Using simpler notation, let:    and   , then    and  
Then,   
In deciding which function to assign to , choose the function that becomes simpler when derivative is
taken, since  is what goes back into the integral as the second component.
When taking the anti-derivative of , constant factor does not need to be added because it is captured
at the end when the whole integral is solved with the addition of the arbitrary constant factor .
Some complex integrals require integration by parts more than once.
For proving reduction formulas, some terms need to be moved to the other side of the equal sign to be
merged with the original integral in order to solve the integral (see example).
Definite Integral
Based on the Fundamental Theorem of Calculus Part 2:



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Description
7.1 Integration Techniques Integration by Parts: Overview Indefinite Integral This follows after the form of the Product Rule: ∫) ( ) ( ) ( ) ∫ ( ) ( ) Using simpler notation, let: ( ) and ( ) , then ( ) and ( ) Then, ∫ ∫ In deciding which function to assign to , choose the function that becomes simpler when derivative is taken, since is what goes back into the integral as the second component. When taking the anti-derivative of , constant factor does not
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