Evaluate the definite integral $\int_{0}^{1} \frac{d x}{1+x^{2}}$.

Let $[t]$ denote the largest integer less than or equal to $t$. If $\int_0^3 \left( [x^2] + [\frac{x^2}{2}] \right) dx = a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7}$,where $a, b, c \in \mathbb{Z}$,then $a + b + c$ is equal to:

The value of the definite integral $\int\limits_2^3 {\left[ {\sqrt {2x - \sqrt {5(4x - 5)} } + \sqrt {2x + \sqrt {5(4x - 5)} } } \right]} \,dx$ is:

If $p(x)$ is a cubic polynomial with $p(1)=3, p(0)=2$ and $p(-1)=4$,then $\int_{-1}^1 p(x) dx$ is

$\int_{\pi /6}^{\pi /4} \text{cosec} \, 2x \, dx = $

The integral $\int_{0}^{2} ||x-1|-x| dx$ is equal to (in $.5$)