Integration of a natural log function: Example 5
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We wish to find
.
We need to express this in terms of a single variable,
w
.
What is a good candidate for replacement with this variable?
Let’s try
to simplify the natural log function.
We also need to determine
dw
. Set up that up.
Determine
dw
.
Note that this combination appears in our problem.
Use these results to convert our problem to one expressed in terms of
w
.
We get
Can we integrate this directly?
No.
Why?
Because we don’t know of a function for which the derivative is
w
*ln(
w
).
Isn’t the integral equal to
?
No, because we would need a factor of
instead of just
dw
in the integral.
How can we solve this problem?
We can try Integration By Parts:
Set that up in general.
What should we take as
u
?
If we let
u
= ln(
w
), then we will get something simpler when we form
du
.
What should we take as
dv
?
As usual,
dv
is the rest of the integrand.
In this case,
dv
=
wdw
.
Determine
du
.
.
Determine
v
.
.
Use these results in Integration by Parts.
Rearrange the first term and cancel in the second.
Do the integration.
Are we done?
No.
What do we do next?
We convert this result to an expression in
x
.
Do that.
We get
Let’s try the same problem by using our
Table Of Integrals
.
Which of the numbered equations in the table does
look like?
It is similar to #2, with an appropriate choice of “
a
”.
What should the value of “
a
” be?
We need
a
= 1.
Do we get the same result?
Yes.
Check this result by differentiation to see if we get our original integrand. Set up the differentiation.
Using the Product Rule in the first term, we get
Do the indicated derivatives.
We get
Cancel and simplify.
We get
which checks.
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General Contents
Detailed Contents
Index