The value of $I = \int_{0}^{1} x \left| x - \frac{1}{2} \right| dx$ is

$\int_{1/4}^{1/2} \frac{dx}{\sqrt{x - x^2}} = $

If $I=\int_0^{\pi / 2} \frac{d x}{5+3 \sin x}=\lambda \tan ^{-1}\left(\frac{1}{2}\right)$, then $\lambda=$

If $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of the integral $\int_0^2 {{x^2}[x]\,dx} $ equals (in $/3$)

The expression $\frac{\int_0^n [x] dx}{\int_0^n \{x\} dx}$, where $[x]$ and $\{x\}$ are respectively the integral and fractional part of $x$ and $n \in N$, is equal to

$\int_0^1 |5x - 3| dx = $