$\int\limits_a^b [x] \,dx + \int\limits_a^b [-x] \,dx$,where $[.]$ denotes the greatest integer function,is equal to:

If $M = \int_{0}^{\pi / 2} \frac{\cos x}{x+2} dx$ and $N = \int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{(x+1)^{2}} dx$, then the value of $M-N$ is

Evaluate the following definite integral: $\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} d x$

$\int_{ - \pi /2}^{\pi /2} {\frac{{\sin x}}{{1 + {{\cos }^2}x}}{e^{ - {{\cos }^2}x}}dx} $ is equal to

If the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{323}}}\right) d x=\frac{\pi}{4}(\pi+a)-2$,then the value of $a$ is

$\int_{-1/2}^{1/2} (\cos x) \left[ \log \left( \frac{1-x}{1+x} \right) \right] dx = $