$\int_0^{b - c} f''(x + a) \, dx = $

The positive value of $x$ satisfying the equation $\int_x^1(1-t) dt = \frac{1}{2}$ is

If $f(t) = \int_{-t}^{t} \frac{dx}{1 + x^2}$,then $f'(1)$ is

$\int_{ - \pi /2}^{\pi /2} {\sqrt {\frac{1}{2}(1 - \cos 2x)} } \,dx = $

Let $[t]$ denote the greatest integer function. If $\int_0^{2.4} [x^2] dx = \alpha + \beta \sqrt{2} + \gamma \sqrt{3} + \delta \sqrt{5}$,then $\alpha + \beta + \gamma + \delta$ is equal to $..............$.

Let $[x]$ be the greatest integer function. Then, $\int_{-1}^{1} [x+2[x+2[x]]] dx = $