Let $f$ be a polynomial function such that $f\left(x^2+1\right)=x^4+5 x^2+2$, for all $x \in \mathbb{R}$.

Then $\int\limits_0^3 f(x) d x$ is equal to

The value of the integral $\int_{\frac{\pi}{24}}^{\frac{5 \pi}{24}} \frac{\mathrm{~d} x}{1+\sqrt[3]{\tan 2 x}}$ is :

The value of $\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right) d x$, where $[\cdot]$ denotes the greatest integer function, is

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function. If $6 \int\limits_1^x f(t) d t=3 x f(x)+x^3-4$ for all $x \geq 1$, then the value of $f(2)-f(3)$ is :