Two Ways of U-Substitution Giving Different Answers

[allimonster]

Hi all!

I've been having some problems with a certain problem that has an integral in it; it seems like there are two ways to substitute u in, and they come out with different answers. Could someone look over my work and see if there is a calculation issue or a concept I'm overlooking that would make one of them incorrect?

This is the original problem:


By defining u as


you can go through the process of u substitution and end up with:


However, I noticed that you could go through the same process but instead use

and substituting that in multiple places:


Is there a technical problem with the second method? The answers are obviously different, but is there a reason to use one over the other? Which one is correct? Or have I just made an error somewhere?

Thank you for your help!

 

[Dr.Peterson]

Check the sign on that last integration you did in the second method. (Write out the details.)

 

[allimonster]

Check the sign on that last integration you did in the second method. (Write out the details.)

Aha! I’m glad it was just a technical error and not something conceptual. Thank you!

 

[Dr.Peterson]

Aha! I’m glad it was just a technical error and not something conceptual. Thank you!

I can add that when I looked at the problem, my initial expectation was that your two answers would turn out to be equivalent, which happens quite commonly in integration. In that case it wouldn't have been an error at all, except in thinking they were different, and likely ignoring the constant. But your answers couldn't have just differed by a constant.