$ \int_{-3}^{3} \cot^{-1} x \, dx = $

Let $a$ be a positive real number such that $\int_{0}^{a} e^{x-[x]} dx = 10e - 9$,where $[x]$ is the greatest integer less than or equal to $x$. Then $a$ is equal to:

If $\int_0^\pi {x\,f({{\cos }^2}x + {{\tan }^4}x)\,dx} = k\int_0^{\pi /2} {f({{\cos }^2}x + {{\tan }^4}x)\,dx,}$ then the value of $k$ is

The value of $\int_{\sqrt{\ln 2}}^{\sqrt{\ln 3}} \frac{x \sin x^2}{\sin x^2 + \sin (\ln 6 - x^2)} dx$ is

Evaluate $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(x) dx$,where $f(x) = \sin |x| + \cos |x|$ for $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.

If $\int_{-\pi / 2}^{\pi / 2} \frac{8 \sqrt{2} \cos x \, dx}{(1+e^{\sin x})(1+\sin ^4 x)} = \alpha \pi + \beta \log _e(3+2 \sqrt{2})$,where $\alpha, \beta$ are integers,then $\alpha^2+\beta^2$ equals.....................