$$\int {{{{{\sin }^8}x - {{\cos }^8}x} \over {1 - 2{{\sin }^2}x{{\cos }^2}x}}dx} $$

$$\int {{2^x}(f'(x) + f(x)\log 2)dx} $$ is

If $\int \frac{\operatorname{cosec}^2 x-2010}{\cos ^{2010} x} d x=-\frac{f(x)}{(g(x))^{2010}}+c$, where $f\left(\frac{\pi}{4}\right)=1$; then the number of solutions of the equation $\frac{f(x)}{g(x)}=\{x\}$ in $[0,2 \pi]$ is/are (where $\{\cdot\}$ represents fractional part function)

$$ \int \frac{\left(\sqrt[3]{x+\sqrt{2-x^2}}\right)\left(\sqrt[6]{1-x \sqrt{2-x^2}}\right)}{\sqrt[3]{1-x^2}} d x ;(x \in(0,1))= $$