Integral of Natural Logarithm
Repeated Use
Solving for the Unknown Integral
Tabular Integration
Integration by Parts
The Product Rule for differentiation says
.Multiplying the equation by
,
we get the differential form
.Integrating both sides of the equation,
or
.Solving for the first integral on the right side, we get the Formula for
Integration by Parts:
.This formula express one integral in terms of a second integral.
The hope is that the second integral will be easier to work out.
See Examples 1 - 4, pages 548 - 550.
Particularly important is Example 4 which gives the Integral of
the Natural Logarithm:
.Repeated Use
At times we must integrate by parts more than once.
Such is the case for the integrals of the form
.See Examples 5, page 550.
Solving for the Unknown Integral
In this case, we integrate by parts, then find again on the right side
of the equation the integral that we started with, so we solve for the
integral to get an equation that gives the solution for the integral.
Such is the case for the integral
.See Example 6, page 551.
Tabular Integration
This is a technique for organizing integration by parts.
See Examples 7 - 8, pages 552 - 553.
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