If $[x]$ is the greatest integer function not greater than $x$,then $\int_0^8 [x] dx$ is equal to

If $\int\limits_0^a {\frac{{dx}}{{\sqrt {x + a} + \sqrt x }}} = \int\limits_0^{\frac{\pi }{8}} {\frac{{2\tan \theta }}{{\sin 2\theta }}} d\theta$,then the value of $a$ is equal to $(a > 0)$.

Evaluate the definite integral $\int_2^5 2[x] \, dx$,where $[x]$ denotes the greatest integer function $\leq x$.

Let $a, b, c$ be non-zero real numbers such that $\int_0^3 {(3ax^2 + 2bx + c)\,dx} = \int_1^3 {(3ax^2 + 2bx + c)\,dx}$,then

If $I_n = \int_0^{\pi/4} \tan^n \theta \, d\theta$ for $n = 1, 2, 3, \ldots$,then $I_{n-1} + I_{n+1}$ 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 $...........$.