<s>[INST] What is indefinite $ \int\frac{\sin\left(x\right)\cos\left(x\right)}{\sqrt{3 - x^{4}}}{\rm d}x . $ $$
\int\frac{\sin\left(x\right)\cos\left(x\right)}{\sqrt{3 - x^{4}}}{\rm d}x .
$$
Hello.
Today, in exam, we had to evaluate this integral. Noone was able to do it. Appreciate any help.
 [/INST] As said in comments, the integral you posted does not seem to be doable.
Let us admit a typo in the problem and let us consider $$I=\int\frac{\sin\left(x\right)\cos\left(x\right)}{\sqrt{3 - \sin^4(x)}}\,{\rm d}x $$ The denominator looking more or less as $\sqrt{1-y^2}$, I thought that $\sin^{-1}(...)$. So, let us try a change of variable such that $$\frac{\sin ^2(x)}{\sqrt{3}}=y$$ that is to say $$x=\sin ^{-1}\left(\sqrt[4]{3} \sqrt{y}\right)$$ $$dx=\frac{\sqrt[4]{3}}{2 \sqrt{y} \sqrt{1-\sqrt{3} y}}$$ $$\sin(x)=\sqrt[4]{3} \sqrt{y}$$ $$\cos(x)=\sqrt{1-\sqrt{3} y}$$ Now, replacing $$I=\frac{1}{2}\int\frac{1}{ \sqrt{1-y^2}}dy=\frac{1}{2} \sin^{-1}(y)=\frac{1}{2} \sin ^{-1}\left(\frac{\sin ^2(x)}{\sqrt{3}}\right)$$
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