How do you know when to use integration by parts with ln?

[calc67x]

Here are some problems that I did:

#1. 1/(xlnx) =ln|ln|x||
#2. lnx/x =ln^2x/2

#3 lnx/sqrtx =2sqrtx*lnx -4sqrtx

I was able to use the u substitution on the first 2 problems, but I had no idea how to approach the 2nd problem until I looked at videos and saw that integration by parts was being used.
How do I know that integration by parts is used on the #3 problem and not on the others without just having lots of experience? Is there a way to tell?
Any ideas are appreciated!

 

[MarkFL]

Here are some problems that I did:

#1. 1/(xlnx) =ln|ln|x||
#2. lnx/x =ln^2x/2

#3 lnx/sqrtx =2sqrtx*lnx -4sqrtx

I was able to use the u substitution on the first 2 problems, but I had no idea how to approach the 2nd problem until I looked at videos and saw that integration by parts was being used.
How do I know that integration by parts is used on the #3 problem and not on the others without just having lots of experience? Is there a way to tell?
Any ideas are appreciated!

If you see an integrand that is a function of both \(\displaystyle \ln(u(x))\) and \(\displaystyle d\frac{1}{u(x)}\), then you know a \(\displaystyle u\) substitution may be used, since:

\(\displaystyle \displaystyle \frac{d}{dx}(\ln(u(x)))=\frac{1}{u(x)}\frac{du}{dx}\)

Otherwise, IBP is something to be considered as a technique for obtaining the anti-derivative.

 

[Dr.Peterson]

Here are some problems that I did:

#1. 1/(xlnx) =ln|ln|x||
#2. lnx/x =ln^2x/2

#3 lnx/sqrtx =2sqrtx*lnx -4sqrtx

I was able to use the u substitution on the first 2 problems, but I had no idea how to approach the 2nd problem until I looked at videos and saw that integration by parts was being used.
How do I know that integration by parts is used on the #3 problem and not on the others without just having lots of experience? Is there a way to tell?
Any ideas are appreciated!

I would just say, if you've tried other things and they didn't work, try parts!

There is often more than one way to integrate; sometimes parts will be the first thing you find that works, while someone else would find a substitution that works. The important thing is just to try things. You will not recognize immediately what works (for the harder problems), even with lots of experience.

Take a look here; there are a number of examples given, showing some of the variety of things that can happen.

Quote: "The general rule of thumb that I use in my classes is that you should use the method that you find easiest. This may not be the method that others find easiest, but that doesn’t make it the wrong method."

Another: "One of the more common mistakes with integration by parts is for people to get too locked into perceived patterns." It will not always be obvious that parts would be a good idea; you don't know until you try it. And sometimes the good choice of u and dv will be quite surprising.

 

[Steven G]

Here are some problems that I did:

#1. 1/(xlnx) =ln|ln|x||
#2. lnx/x =ln^2x/2

#3 lnx/sqrtx =2sqrtx*lnx -4sqrtx

I was able to use the u substitution on the first 2 problems, but I had no idea how to approach the 2nd problem until I looked at videos and saw that integration by parts was being used.
How do I know that integration by parts is used on the #3 problem and not on the others without just having lots of experience? Is there a way to tell?
Any ideas are appreciated!

What makes you think that any of these equals signs below are valid????
1/(xlnx) = ln|ln|x||
lnx/x = ln^2x/2
lnx/sqrtx = 2sqrtx*lnx -4sqrtx

 

[calc67x]

Thanks, Really helpful!

Thanks, Dr. Peterson for this link. I will check it out. It is good to know that there is no hard and fast rule that I missed!
I will spend more time reviewing various problems to see how they can be done.

I would just say, if you've tried other things and they didn't work, try parts!

There is often more than one way to integrate; sometimes parts will be the first thing you find that works, while someone else would find a substitution that works. The important thing is just to try things. You will not recognize immediately what works (for the harder problems), even with lots of experience.

Take a look here; there are a number of examples given, showing some of the variety of things that can happen.

Quote: "The general rule of thumb that I use in my classes is that you should use the method that you find easiest. This may not be the method that others find easiest, but that doesn’t make it the wrong method."

Another: "One of the more common mistakes with integration by parts is for people to get too locked into perceived patterns." It will not always be obvious that parts would be a good idea; you don't know until you try it. And sometimes the good choice of u and dv will be quite surprising.