[itex]\int\frac{n}{(n^{2}+1)^{2}}[/itex]= itself w/ Partial Fractions

[Lebombo]

Homework Statement

Why when I try to evaluate this with Partial Fractions, why do I end up with the original function?
$\int\frac{n}{(n^{2}+1)^{2}}$

$\frac{n}{(n^{2}+1)(n^{2}+1)}$

$\frac{Ax+B}{n^{2}+1} + \frac{Cx+D}{(n^{2}+1)^{2}}$

1n = (An+B)(n^{2}+1) + Cx + D

$0n^{3}+ 0n^{2} + 1n + 0n^{0} = n^{3}(A) + n^{2}(B) + n(A+C) + n^{0}(B+D)$

A=0 B=0 C=1 D=0

$\frac{0n+0}{n^{2}+1} + \frac{1n+0}{(n^{2}+1)^{2}}$

= 0 + $\frac{n}{(n^{2}+1)^{2}}$

 

[UltrafastPED]

It's already in final form, so C=1 and the rest are 0.

See http://tutorial.math.lamar.edu/Classes/CalcII/PartialFractions.aspx

 

[Lebombo]

That website looks all scrambled up. Nothing seems to be aligned. Which part pertains to why this particular function ends up to simply be itself?

By the way, when I do u-sub on this function I get 1/4 and when I do partial fractions I get -1/4. Is that a sign mistake?

Also, the basis for this question was actually in regards to the series of the function in question.

I was trying to do partial fractions to create a telescoping function to evaluate the series. But partial fractions doesn't separate this function into telescoping series.

So my follow up questions is: Why does this function not separate into a telescoping form if Partial fractions can be applied to it?

 

[Mark44]

Lebombo,
What you're saying doesn't make a lot of sense. As you've seen, partial fractions leads you right back to the exact same integrand, so I don't see how you were able to carry out the integration that way. Substitution is the way to go here.

The integral looks like it would be amenable to a trig substitution, but that seems like a lot of work when an ordinary substitution would do the trick.

 

[UltrafastPED]

That website looks all scrambled up. Nothing seems to be aligned. Which part pertains to why this particular function ends up to simply be itself?

The site works fine with Chrome on my laptop. Check your browser!

The website page goes over the method in detail, with a few examples. A review of the method may help you to find any problems with your technique.


Wolfram Alpha says: $n^2/(n^2+1) = 1-1/(n^2+1)$

 

[SteamKing]

IDK why you want to use partial fractions anyway. You've plainly got a case for a u substitution, no partial fractions or trig substitution required.

 

[Saitama]

I don't see the need of a trig substitution. There is a very obvious substitution you can use as SteamKing has said.

 

[Lebombo]

I just [thought I] saw an opportunity to practice a partial fraction from the past chapter while working on series.

Thanks all.