$\mathop {\lim }\limits_{n \to \infty } \left[ {\frac{1}{{{n^2}}}{{\sec }^2}\frac{1}{{{n^2}}} + \frac{2}{{{n^2}}}{{\sec }^2}\frac{4}{{{n^2}}} + ..... + \frac{1}{n}{{\sec }^2}1} \right]$ equals

For $0 < x < \frac{\pi}{2}$,the integral $\int_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} \ln(e^{\cos x}) \, d(\sin x)$ is equal to:

Let the function $f :[0,2] \rightarrow R$ be defined as $f(x)=\begin{cases} e^{\min \{x^2, x-[x]\}}, & x \in[0,1) \\ e^{[x-\log_e x]}, & x \in[1,2] \end{cases}$ where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int_0^2 x f(x) dx$ is

If $f(x) = \begin{cases} \sqrt{1 - x} & 0 \le x \le 1 \\ (7x - 6)^{-1/3} & 1 < x \le 2 \end{cases}$,then $\int_{0}^{2} f(x) \, dx$ is equal to

Let $f_n = \int_0^{\frac{\pi}{2}} \left(\sum_{k=1}^n \sin^{k-1} x\right) \left(\sum_{k=1}^n (2k-1) \sin^{k-1} x\right) \cos x \, dx$,where $n \in N$. Then $f_{21} - f_{20}$ is equal to $...........$.

If $\int_{n}^{n+1} f(x) dx = n^2 + n$ for all $n \in I$,then the value of $\int_{-3}^{3} f(x) dx$ is equal to