$$\int x \sqrt{\frac{2 \sin \left(x^2+1\right)-\sin 2\left(x^2+1\right)}{2 \sin \left(x^2+1\right)+\sin 2\left(x^2+1\right)}} d x=$$

If $$\int \frac{\cos 8 x+1}{\cot 2 x-\tan 2 x} \mathrm{~d} x=\mathrm{A} \cos 8 x+\mathrm{c}$$, where $$\mathrm{c}$$ is an arbitrary constant, then the value of $$\mathrm{A}$$ is

The value of $$\int(1-\cos x) \cdot \operatorname{cosec}^2 x d x$$ is

If $$\mathrm{I}=\int \sin (\log (x)) \mathrm{d} x$$, then $$\mathrm{I}$$ is given by