Evaluate $\int \frac {x\sin\left(x\right) }{ \sqrt{3 + \sin^{2}\left(x\right)}}\; \operatorname dx$ Evaluate the indefinite integral
$$
\int \dfrac {x\sin\left(x\right) }{ \sqrt{3 + \sin^{2}\left(x\right)}}\; \operatorname dx
$$

I think it's hard work to solve and I don't have any idea. So please give an idea to solve this problem, and I will think about it, thanks.


$$3+\sin^2x=4-\cos^2x\;\implies \int\frac{\sin x\,dx}{\sqrt{4-\cos^2x}}=$$
$$=\int\frac{\frac{\sin x}2\,dx}{\sqrt{1-\left(\frac{\cos x}2\right)^2}}$$
And now use:
Integration by parts:
$$u=x\;,\;\;v'=\frac{\sin}{\sqrt{3+\sin^2x}}$$
Almost immediate integrals:
$$\int\frac{f'(x)dx}{\sqrt{1-f^2(x)}}=\arcsin f(x)+C$$
