If $I = \int_0^\pi x \left\{ \sin^2(\sin x) + \cos^2(\cos x) \right\} dx$,then $[I] = \ldots$. Here,$[.]$ denotes the greatest integer function.

$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left( \frac{x+\frac{\pi}{4}}{2-\cos 2x} \right) dx$ is equal to

$\int_0^1 \frac{\log _e(1+x)}{1+x^2} d x=$

The value of $\int_{0}^{4042} \frac{\sqrt{x} \, dx}{\sqrt{x}+\sqrt{4042-x}}$ is equal to

If $m, l, r, s, n$ are integers such that $9 > m > l > s > n > r > 2$ and $\int_{-2 \pi}^{2 \pi} \sin ^m x \cos ^n x \, dx = 4 \int_0^\pi \sin ^m x \cos ^n x \, dx$, $\int_{-\pi}^\pi \sin ^r x \cos ^s x \, dx = 4 \int_0^{\pi / 2} \sin ^r x \cos ^s x \, dx$ and $\int_{-\pi / 2}^{\pi / 2} \sin ^l x \cos ^m x \, dx = 0$, then which of the following is true?

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 ........ .