Can`t find the integral $\int\frac{dx}{\sqrt x\sin(x)}$

Question

$\int\frac{dx}{\sqrt x\sin(x)}$ 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 $v=\int{\frac{e^{-\int{p(x)\,dx}}}{\sin(x)/\sqrt(x)}}\,dx$.

Answer 1

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.

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

Answer 2

The solution you have suggests that you let $$ y(x)=v(x)/\sqrt{x}. $$ Doing so, your differential equation will turn into $$ x^{3/2}(v''(x)+v(x))=0. $$ I'm sure you can take it from there.

(I don't see where you got your integral from, so I cannot help you with that.)