Existence/Limit of Improper Integral I got the improper integral: 
$$ \int_0^\infty \frac{x^2}{x^4+1} \, dx $$
On one hand one could of course just evaluate the integral and then apply limits.
However, it's not always practical to find an antiderivative.
I got a hint that one can trace back improper integrals to sequences and thus just calculate the limit of a sequence which is way easier, could someone explain to me how this works or link me to some theorems?
Thanks.

$$I=\int_{0}^{1}\frac{x^2}{x^4+1}\,dx + \int_{1}^{+\infty}\frac{x^2}{x^4+1}\,dx = \int_{0}^{1}\left(\frac{x^2}{x^4+1}+\frac{1}{x^4+1}\right)\,dx \tag{1}$$
leads to:
$$ I = \int_{0}^{1}\frac{1+x^2-x^4-x^6}{1-x^8}\,dx = \sum_{k\geq 0}\left(\frac{1}{8k+1}+\frac{1}{8k+3}-\frac{1}{8k+5}-\frac{1}{8k+7}\right)\tag{2}$$
but due to the digamma reflection formula we have:
$$ \sum_{k\geq 0}\left(\frac{1}{8k+1}-\frac{1}{8k+7}\right)=\frac{\pi}{8}\cot\frac{\pi}{8}, \quad \sum_{k\geq 0}\left(\frac{1}{8k+3}-\frac{1}{8k+5}\right)=\frac{\pi}{8}\cot\frac{3\pi}{8}\tag{3}$$
hence it follows that:
$$ I = \frac{\pi}{8}\left[\cot\frac{\pi}{8}+\cot\frac{3\pi}{8}\right]=\color{red}{\frac{\pi}{2\sqrt{2}}}.\tag{4}$$
The same can be achieved by partial fraction decomposition, since $x^4+1=\Phi_8(x)=(x^2-x\sqrt{2}+1)(x^2+x\sqrt{2}+1)$.
Still another way is to apply Glasser's master theorem:
$$ \int_{0}^{+\infty}\frac{dx}{x^2+\frac{1}{x^2}} = \int_{0}^{+\infty}\frac{dx}{\left(x-\frac{1}{x}\right)^2+2}=\int_{0}^{+\infty}\frac{dx}{x^2+2}.\tag{5} $$
