Quote by shubox
1. The problem statement, all variables and given/known data
∫(dx/((1+x^2)^2) from 0 to ∞
Determine whether the improper integral converges and if so, evaluate it.
2. Relevant equations
1+ tan^2(x) = sec^2(x)
1/sec(x) = cos(x)
3. The attempt at a solution
Initially I had no idea how to approach this problem. The answer in the back of the book is ∏/4, which tells me that maybe trig integrals are involved. So i started off with:
lim(R→∞) ∫(dx/((1+x^2)^2) from 0 to R.
x=tan(x)
lim(R→∞) ∫(dx/((1+tan^2(x))^2) from 0 to R.
=lim(R→∞) ∫(dx/(sec^4(x)) from 0 to R.
I do not know where to go from here. Any help would be appreciated

It's not a good idea to use the same variable in your substitution.
Let x=tan(θ) then dx = sec
^{2}(θ) dθ .
After finding the antiderivative in terms of θ, change back to x, plug in the limits of integration, then take the limit.