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Programmed tutorial: Integration of a natural log function: Example 2
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We wish to find
Can we do this directly?
General Contents
Detailed Contents
Index
No.
Why?
Because we don’t know of a function for which the derivative is x*ln(3
x
).
Isn’t the integral equal to
No, because we would need a factor of 3
dx/x
instead of just
xdx
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(3
x
), 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
=
x*dx
.
Determine
du
.
Determine
v
.
Use these results in Integration by Parts.
Rearrange the first term and cancel in the second.
Do the integration.
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
= 3.
Do we get the same result?
Yes.
Check this result by differentiation to see if we get our original integrand.
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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