If $\displaystyle\int \dfrac{dx}{x^{7/2}(x^4 + 1)^{3/8}} = m\left(\dfrac{x^4 + 1}{x^4}\right)^n + c$, where $c$ is a constant of integration, then the value of $\dfrac{n}{m}$ is...

If $[x]$ is the greatest integer function not greater than $x$, then the value of $\displaystyle\int_0^2 x[x^2]\,dx$ is...

The value of $\displaystyle\int_2^4 (\{x\} + [x])\,dx =$...(where $\{x\}$ and $[x]$ are the fractional part function and the greatest integer function, respectively)

If the area of the region bounded by the parabola $y^2 = 4kx$ and the line $x = k$, (where $k > 0$) is $\dfrac{128}{3}$ sq. units, then the value of $\sin^{-1}\left(\dfrac{2}{k}\right)$ is equal to...