Evaluate $\int_\pi^\infty \frac{\sin x}{\ln x}$

Question

Evaluate $\int_\pi^\infty \frac{\sin x}{\ln x}$.

I tried to use THE FEYNMAN WAY, but it doesn't help.

Answer 1

As @Sputnik commented, introducing the factor $e^{-t\ln x}$ and differentiating with respect to $t$ gives $$\int^\infty_\pi\frac{\sin x}{x^t}dx$$ By the substitution $u=x-\pi$, you get $$-\int^\infty_0(u+\pi)^{-t}\sin(u)du$$

Applying the generalized binomial theorem, the integral equals $$-\sum^{\infty}_{k=0}\frac{(-t)_k}{k!}\pi^{k}\int^\infty_0u^{-t-k}\sin (u)du$$

As mentioned in my previous post, this equals $$-\sum^\infty_{k=0} \frac{(-t)_k}{k!}\pi^{k}\cos(\frac{\pi(t+k)}2)\Gamma (1-t-k)$$

I wonder if one can further simplify it. Moreover, don’t forget to integrate it with respect to $t$ and find the constant!