Evaluate the definite integral $\int_{-1}^{2} \left[ \frac{[x]}{1 + x^2} \right] dx$,where $[\cdot]$ denotes the Greatest Integer Function $(GIF)$.

Let the domain of the function $f(x) = \log_2 \log_4 \log_6(3 + 4x - x^2)$ be $(a, b)$. If $\int_0^{b-a} [x^2] dx = p - \sqrt{q} - \sqrt{r}$,where $p, q, r \in \mathbb{N}$,$\gcd(p, q, r) = 1$,and $[\cdot]$ is the greatest integer function,then $p + q + r$ is equal to

Let $n$ be a positive integer. For a real number $x$,let $[x]$ denote the largest integer not exceeding $x$ and $\{x\}=x-[x]$. Then,$\int \limits_1^{n+1} \frac{(\{x\})^{[x]}}{[x]} d x$ is equal to

$\int_1^2 \frac{1}{x^2} e^{-\frac{1}{x}} \, dx = $

$\int_0^{\pi /2} {\frac{{\sin x\cos x\,dx}}{{{{\cos }^2}x + 3\cos x + 2}}} = $

The function $L(x) = \int_1^x \frac{dt}{t}$ satisfies the equation