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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