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:

$$\int\frac{f'(x)dx}{\sqrt{1-f^2(x)}}=\arcsin f(x)+C$$

• Unless you write the full answer, I don't see how is the $x$-factor involved in your answer. – Felix Marin Jan 26 '14 at 7:03
• @FelixMarin $$u=x\;\;,\;\;u'=1\\v'=-\frac{-\sin x}{\sqrt{1+\sin^2}}=-\frac{-\frac{\sin x}2}{\sqrt{1-\left(\frac{\cos x}2\right)^2}}\;\;,\;\;v=\arcsin\frac{\cos x}2$$ – DonAntonio Jan 26 '14 at 7:05