Can`t find the integral $\int\frac{dx}{\sqrt x\sin(x)}$ $\int\frac{dx}{\sqrt x\sin(x)}$ Im stuck in this problem for about 40 minutes. Any clue how to solve this?
The actual problem is an ODE problem, it is given $x^2y''+xy'+(x^2-1/4)y=0$ and one of the solutions is $y_1(x)=\sin(x)/\sqrt(x)$. So I need to find $v=\int{\frac{e^{-\int{p(x)\,dx}}}{\sin(x)/\sqrt(x)}}\,dx$.

With $x=t^2$ one gets $\displaystyle \int \frac{dx}{\sqrt{x}\sin x}=2\int \frac{dt}{\sin t^2}=2(-\frac{1}{t}+\frac{t^3}{18}+\frac{t^7}{360}+O(t^9))+C$ . 
I think there is no elementary solution. 

Note: $\enspace$ Perhaps you need $\displaystyle \int \frac{dx}{\sin x}=-\ln(\frac{1+\cos x}{\sin x})+C \,\,$ ?
