Let $f(x)=\int^{\infty}_{-\infty}\frac{1}{x^4+1}dx$
I am required to prove that the integral exists (i.e. NOT CALCULATE IT!)
What am I supposed to use? Normally to prove the existence of an improper integral I would use the integral comparison theorem (not sure on the name in english) which states:
$\int^{\infty}_{1}f(x)dx$ converges iff $\sum^{\infty}_{n=1} f(n)$ converges. However this only pertains to the lower bound being 1?!
Any help is greatly appreciated!
If $\int_1^\infty$ exists and (by symmetry) $\int_{-\infty}^{-1}$ exists, and $\int_{-1}^1$ poses no problem at all, then the whole integral exists.