(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

It's been a couple of years since I've done real math, so I'm kinda stuck on this one. This is actually part of a physics problem, not a math problem - but I'm stuck on the calculus part. I'm trying to solve this guy:

[itex]

\int \limits_{-\infty}^{\infty} \frac{x^2}{(x^2+a^2)^2}\textrm{d}x

[/itex]

a is a constant

2. Relevant equations

[itex]\textrm{tan}^2 \theta + 1 = \textrm{sec}^2 \theta[/itex]

3. The attempt at a solution

I make the substitution

[itex] x = a \textrm{tan} \theta[/itex]

therefore

[itex]

\textrm{d}x = a\textrm{sec}^2\theta\textrm{d}\theta

[/itex]

giving me

[itex]

\int{\frac{a^2 \textrm{tan}^2 \theta a\textrm{sec}^2 \theta \textrm{d} \theta}{(a^2 \textrm{tan}^2 \theta + a^2)^2}}

[/itex]

and eventually I get it to boil down to (using the aforementioned tangent identity and canceling terms)

[itex] \frac{1}{a} \int \textrm{tan}^2 \theta \textrm{d} \theta [/itex]

I thought I was supposed to change the limits to

[itex] \pm\frac{\pi}{2} [/itex]

, but when I solve the above simplified integral I get

[itex] \textrm{tan}\theta - \theta[/itex]

which is not convergent

My problem is taking the limit for the tangent at [itex]\pm\frac{\pi}{2}[/itex]

I'm probably screwing up with the limits of integration. What exactly am I supposed to do with a trig substitution and the limits when dealing with an improper integral? I was following an old calculus book of mine, but this doesn't seem exactly right...

Thanks for your help!

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# Improper integrals and trig substitution

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