How did you prove this integral?
Solution 1:
The answer seems not to be true. Here are the arguments. Let $x=\pi/4$, then the denominator is a differentiable function of $t$ on $[0.05,1]$ as the series $$ \sum_{k=1}^\infty e^{-0.05k^2}$$ rapidly converges. Up to a numerical solution, the denominator equals zero at $t=0.08560978493249566136215339$ and the numerator is positive on $[0.05,1].$ Therefore, the integral under consideration diverges in the case $x=\pi/4.$ See the calculations in a Maple file exported as a *.pdf file which can be downloaded from RapidShare.