Let for some function $y=f(x)$,$\int_0^x t f(t) d t=x^2 f(x)$,$x > 0$ and $f(2)=3$. Then $f(6)$ is equal to :

$\int_{-5}^5 x^4\left(25-x^2\right)^{5 / 2} d x=$

$\int_{-\pi}^\pi \frac{\cos ^{2022} x}{1+(2022)^x} d x=$

$\int_0^\pi (\sin^3 x + \cos^2 x)^2 dx = $

$\mathop {Lim}\limits_{k \to 0} \frac{1}{k} \int\limits_0^k (1 + \sin 2x)^{\frac{1}{x}} dx$

Let $f : R \rightarrow R$ be a continuous odd function,which vanishes exactly at one point and $f(1) = \frac{1}{2}$. Suppose that $F(x) = \int_{-1}^x f(t) dt$ for all $x \in [-1, 2]$ and $G(x) = \int_{-1}^x t|f(f(t))| dt$ for all $x \in [-1, 2]$. If $\lim_{x \rightarrow 1} \frac{F(x)}{G(x)} = \frac{1}{14}$,then the value of $f\left(\frac{1}{2}\right)$ is