Does $\displaystyle \int_{0}^{\infty} \frac{\sin(\tan x)}{x} \ dx $ converge?
A plot of the integrand shows that it oscillates wildly near $\displaystyle x= \frac{\pi}{2} + \pi n$ $.
Does $\displaystyle \int_{0}^{\infty} \frac{\sin(\tan x)}{x} \ dx $ converge?
A plot of the integrand shows that it oscillates wildly near $\displaystyle x= \frac{\pi}{2} + \pi n$ $.
Would integrating by parts allow the integral of the periodic function $\displaystyle \sin{\tan{x}}$ to be rewritten as a sum? Then you could prove the sum diverges and therefore the integral diverges too. Haven't tried but it seems it would work.