Why is this integral not absolutely integrable? I would like to prove that the following integral is not absolutely convergent:
$$\int\limits_{1}^\infty \frac{1}{x}\cdot \sin(\alpha x+c)\cdot \sin(\beta x+ c)dx.$$
$\alpha,\beta,c>0$ are real constants. I could manage to prove that this integral is convergent for $\alpha\neq \beta$. But I do not know why it is not absolutely convergent. Can anyone give me a hint?
Any help will be very appreciated.
Best wishes
 A: If $\alpha=\beta$ the integral is not even conditionally convergent, hence we may assume $\alpha\neq\beta$. 
By Fourier series or other means it is not difficult to show that for some $K_{\alpha,\beta}>0$
$$ \int_{1}^{N}\left|\sin(\alpha x+c)\right|\cdot \left|\sin(\beta x+c)\right|\,dx = K_{\alpha,\beta} N + O(1)\tag{1} $$
as $N\to +\infty$, hence by integration by parts
$$ \int_{1}^{N}\frac{\left|\sin(\alpha x+c)\right|\cdot \left|\sin(\beta x+c)\right|}{x}\,dx =O(1)+K_{\alpha,\beta}\int_{1}^{N}\frac{dx}{x}\gg\log(N)\tag{2}$$
and the integral in the LHS is divergent as $N\to +\infty$.
It is also possible to prove that as soon as $\alpha\neq\beta$,
$$ \int_{1}^{N}\frac{\left|\sin(\alpha x+c)\right|\cdot \left|\sin(\beta x+c)\right|}{x}\,dx = \frac{4}{\pi^2}\log(N)+O(1)\tag{3} $$
details are given in the comment below: in particular, the constant $K_{\alpha,\beta}$ appearing in $(1)$ does not really depend on $\alpha$ or $\beta$, it is just $\frac{4}{\pi^2}$, and $(3)$ follows by summation by parts.
