Evaluate the integral: $\int_{-1}^{1} \left[ \sqrt{1+x+x^{2}} - \sqrt{1-x+x^{2}} \right] dx$

The true set of values of $a$ for which the inequality $\int_{a}^{0} (3^{-2x} - 2 \cdot 3^{-x}) \, dx \geq 0$ is true is:

If the integral $\int_{0}^{10} \frac{[\sin 2 \pi x ]}{ e ^{ x -[ x ]}} dx =\alpha e ^{-1}+\beta e ^{-\frac{1}{2}}+\gamma$,where $\alpha, \beta, \gamma$ are integers and $[ x ]$ denotes the greatest integer less than or equal to $x$,then the value of $\alpha+\beta+\gamma$ is equal to ........ .

The value of the integral $\int_{4}^{10} \frac{[x^2]}{[(x-14)^2] + [x^2]} dx$,where $[x]$ denotes the greatest integer function,is

If $f(a+b-x)=f(x)$,then $\int_a^b x f(x) d x=$ . . . . . . .

If $M=\int_0^{\infty} \frac{\log t}{1+t^3} d t$ and $N=\int_{-\infty}^{\infty} \frac{t e^{2 t}}{1+e^{3 t}} d t$,then