OpenStax-CNX module: m47412
1
∗
Integral of x*ln(3x) by Parts
John Taylor
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0†
Abstract
Methods for the integral of x*ln(3x) are discussed. Integration by Parts is demonstrated and checked
by dierentiation.
1 Integration of x*ln(3x) by parts
(The small steps below can be used for a self test. To do so, Scroll in small increments.)
We wish to nd
Can we do this directly?
No.
Why?
Because we don't know of a function for which the derivative is x*ln(3x ).
Isn't the integral equal to
No, because we would need a factor of dx/x instead of 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(3x ), 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.
∗ Version
1.3: Sep 2, 2013 9:27 am -0500
† http://creativecommons.org/licenses/by/3.0/
http://cnx.org/content/m47412/1.3/
OpenStax-CNX module: m47412
2
Determine v.
Use these results in Integration by Parts.
Rearrange the rst term and cancel in the second.
Do the integration.
Check this result by dierentiation to see if we get our original integrand.
Using the Product Rule in the rst term, we get
Do the indicated derivatives.
We get
Cancel and simplify.
We get
,
which checks.
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