integral of reciprocal function

[spiral birch]

hello! i was doing this question - "find the integral of x/(1-x^2) between 0 and 2" and got quite confused.

at first glance, the asymptote at x=1 led me to think the integral is divergent/"infinite" - however, using an alternative method, I also ended up getting -ln(3)/2? From a graph sketch, the negative area > the positive area, so that would make sense, but I'm still not 100% sure.

please do critique my horrible workings out + put me out of my misery lol

[south scroll]

👋 Welcome to your Help Thread!

Hey @spiral birch, Thanks for sharing your math problem with us. While you wait for a Helper to help you, we want to share some vital information with you.

​

● Please take a moment to read the helpee guidelines. This will make sure that your post follows the helpee rules of our community.

​

● Please don't ping <@&775784618955505685>, <@&1283689826742440016>, <@&819616364188139550>, or <@&624327278137966593> for help because their job is to take care of the server's administrative tasks, not to answer queries directly. However, if you have a problem with how a Helper is acting, you can ping a Helper Moderator.

​

● It's always very useful if you can show us the work you've done so far. This makes it easier for our Helpers to find mistakes and help you get to the right answer.

Once you've got your answer and you're all set, you can close this thread by typing +close, and then you can click the Thank buttons to show your Helper some appreciation.

[spiral birch]

i do realise this question in particular is probably way way easier than most here

[merry scarab]

hi

[merry scarab]

u r not supposed to apply limit seperately there

like lim (n tends to infinity ) 1/n^2 + 2/n^2 +3/n^2 ... n/n^2

you cant just assume all the terms to be 0 as n is going to infiity

i.e.

u can not apply limit seperately

if u apply it correctly then the + intinfity and - infitny will resolve into something

[spiral birch]

ohh i see - thanks! ill continue with this soon + hopefully get something

[drifting cargo]

You can't use Newton-Leibnitz like this when the antiderivative you use is discontinuous.
SInce the function behaves like -1/(2(x - 1)) around x = 1, the integral diverges.

[spiral birch]

ah gotcha - yea that makes sense
so therefore there is no (for lack of a better term) "proper" answer?

[drifting cargo]

Well, the answer is that the integral diverges.
What you did is more similar to this: https://en.wikipedia.org/wiki/Cauchy_principal_value

But by default that isn't done.

[spiral birch]

ooh ok nice - thank you! 🙏

[south scroll]

@spiral birch

:HelpIcon:| Help Reminder

Hello diacr1tik, this is a friendly reminder that your help request has been inactive for more than 24 hours. If you no longer need assistance, please consider closing the thread using the +close command. This thread will be automatically closed in 3 days if it remains inactive.