# Math Help - Improrper Integrals

1. ## Improrper Integrals

I have a midterm tomorrow morning and was looking at example problems and got stuck on this one. Please Help!

Determine if the following integral converges or diverges. If it converges find its value
It is the integral from 8 to +infinity of
[(64-x^2)/(64+x^2)^2]*arctan(x/8) dx
It tells us that the integral of [(64-x^2)/(64+x^2)^2] = x/(64+x^2)dx

I know that since they give us the integral for half of the function to use integration by parts. I have gotten to the point where you plug the infinities back into the integral.

for my integral I got arctan(x/8)*x/(64+x^2)+4/(64+x^2) evaluated from 8 to +inf

I have no problem plugging 8 into the integral but I am having trouble plugging the +inf into the integral and knowing whether or not it is determinate or indeterminate

2. You have

$\displaystyle\int_{8}^{\infty}\frac{64-x^{2}}{64+x^{2}}\,\tan^{-1}(x/8)\,dx.$

$dv=\frac{64-x^{2}}{64+x^{2}}\,dx,$ and let
$u=\tan^{-1}(x/8).$ Then the integral becomes
$\displaystyle\int_{8}^{\infty}\frac{64-x^{2}}{64+x^{2}}\,\tan^{-1}(x/8)\,dx=\left(\tan^{-1}(x/8)\,\frac{x}{64+x^{2}}\right)\Bigg|_{8}^{\infty}-\int_{8}^{\infty}\frac{x}{64+x^{2}}\,\frac{8}{64+x ^{2}}\,dx$
$\displaystyle=\left(\tan^{-1}(x/8)\,\frac{x}{64+x^{2}}\right)\Bigg|_{8}^{\infty}-8\int_{8}^{\infty}\frac{x}{(64+x^{2})^{2}}\,dx.$