If $$\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^2 \theta} d \theta=A \log _e|f(\theta)|+c$$ (where $$c$$ is a constant of integration), then $$\frac{f(\theta)}{A}$$ can be

$$\int \frac{\sin x+\sin ^3 x}{\cos 2 x} d x=A \cos x+B \log \mathrm{f}(x)+c$$ (where $$\mathrm{c}$$ is a constant of integration). Then values of $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{f}(x)$$ are

If $$\int \frac{x^3 \mathrm{~d} x}{\sqrt{1+x^2}}=\mathrm{a}\left(1+x^2\right) \sqrt{1+x^2}+\mathrm{b} \sqrt{1+x^2}+\mathrm{c}$$ (where $$\mathrm{c}$$ is a constant of integration), then the value of $$3 \mathrm{ab}$$ is

Let $$f(x)=\int \frac{x^2-3 x+2}{x^4+1} \mathrm{~d} x$$, then function decreases in the interval