If $f(x) = \frac{2 - x \cos x}{2 + x \cos x}$ and $g(x) = \ln x$ for $x > 0$,then the value of the integral $\int_{-\pi/4}^{\pi/4} g(f(x)) dx$ is

If $f(x) = f(2 - x)$,then $\int_{0.5}^{1.5} xf(x) dx$ equals

$\int_0^\pi x \sin^3 x \, dx = $

The value of $\int_{0}^{\frac{\pi}{2}} \ln \left(\frac{4+3 \sin x}{4+3 \cos x}\right) d x$ is

The number of continuous functions $f:[0,1] \rightarrow [0,1]$ such that $f(x) < x^2$ for all $x \in (0,1]$ and $\int_{0}^{1} f(x) dx = \frac{1}{3}$ is:

Let $f$ be a non-constant continuous function for all $x \geq 0$. Let $f$ satisfy the relation $f(x) f(a-x)=1$ for some $a \in R^{+}$. Then, $I=\int_{0}^{a} \frac{d x}{1+f(x)}$ is equal to