SOLUTION: Hi, I was wondering if someone could help me with this question.
Evaluate the following indefinite integral using integration by parts:
ln(x)/x^(3) dx
Thanks, Judy
Algebra.Com
Question 226112: Hi, I was wondering if someone could help me with this question.
Evaluate the following indefinite integral using integration by parts:
ln(x)/x^(3) dx
Thanks, Judy
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
I'm going to presume you want to evaluate:
}{x^3}\,dx})
Using integration by parts. First split the integrand into two parts, 
and 
. Select the parts so that when you differentiate you get something simpler and when you integrate you get something simpler. The split here should be )
because it differentiates to 
and 
because it integrates to 
So:
)
therefore
and
therefore
Now use the Integration by Parts Formula:
Substituting:
}{x^3}\,dx}\ =\ -\frac{\ln(x)}{2x^2}\ -\ \int{\left(\frac{-1}{2x^2}\right)\left(\frac{1}{x}\right)\,dx)
\left(\frac{1}{x}\right)\,dx\ =\ \int{\frac{-1}{2x^3}\,dx\ =\ \frac{1}{4x^2}\ +\ C)
, so:
}{x^3}\,dx}\ =\ -\frac{\ln(x)}{2x^2}\ -\ \frac{1}{4x^2}\ +\ C)
Which still can be simplified somewhat, but I think you get the idea.
John
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