zje
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Homework Statement
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:
<br /> \int \limits_{-\infty}^{\infty} \frac{x^2}{(x^2+a^2)^2}\textrm{d}x<br />
a is a constant
Homework Equations
\textrm{tan}^2 \theta + 1 = \textrm{sec}^2 \theta
The Attempt at a Solution
I make the substitution
x = a \textrm{tan} \theta
therefore
<br /> \textrm{d}x = a\textrm{sec}^2\theta\textrm{d}\theta<br />
giving me
<br /> \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}}<br />
and eventually I get it to boil down to (using the aforementioned tangent identity and canceling terms)
\frac{1}{a} \int \textrm{tan}^2 \theta \textrm{d} \theta
I thought I was supposed to change the limits to
\pm\frac{\pi}{2}
, but when I solve the above simplified integral I get
\textrm{tan}\theta - \theta
which is not convergent
My problem is taking the limit for the tangent at \pm\frac{\pi}{2}
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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