The value of $\int_{0}^{\sqrt{2}} [x^2] \, dx$,where $[.]$ denotes the greatest integer function.

$\int_0^1 (1+x) \log (1+x) \, dx =$

Evaluate the following definite integral as the limit of a sum:
$\int_{1}^{4}(x^{2}-x) dx$

Let $g(x) = \int_0^x f(t) \, dt$ where $\frac{1}{2} \le f(t) \le 1$ for $t \in [0, 1]$ and $0 \le f(t) \le \frac{1}{2}$ for $t \in (1, 2]$. Then which of the following is true for $g(2)$?

$\int_1^5 (|x-3| + |1-x|) \, dx =$

The value of the integral $\int_{\pi/6}^{\pi/3} \left(\frac{4 - \csc^2 x}{\cos^4 x}\right) dx$ is: