Given $$\int_{0}^{\infty}\frac{\cos(\ln x)}{\sqrt{x^{4}+1}}\text dx$$
I need to check if the integral absolutely converges, Conditionally converges or diverges.
what I did was to use Linearity, given $$\int_{1}^{\infty}\frac{\cos(\ln x)}{\sqrt{x^{4}+1}}\text dx$$
I've tried to make u-substitution as: $x^2 = t$ and got $$\frac{1}{2}\int_{1}^{\infty}\frac{\cos(\frac{1}{2}\ln t))}{\sqrt{t^{3}+t}}\text dx$$
than I tried to make another u-substitution as: $\ln t = k$ so that: $$\frac{1}{2}\int_{1}^{\infty}\frac{\cos(\frac{k}{2})\exp^{k}}{\sqrt{\exp^{3k}+\exp^{k}}}\text dx$$
but on that point I got stuck as another substitution would not help and also Integration by parts will not work.