$\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}$ is equal to :

For $n \in N$,if $I_n = \int \frac{\sin nx}{\sin x} dx = \frac{2}{n-1} \sin(n-1)x + I_{n-2}$ and $\int_0^\pi \frac{\sin nx}{\sin x} dx = \frac{k\pi}{2}$,then $k =$

If ${I_n} = \int_0^{\pi /4} {{\tan ^n}\theta \,d\theta }$,then ${I_8} + {I_6}$ equals

The value of the integral $\int_{-2}^{2}(1+2 \sin x) e^{|x|} d x$ is equal to

$\int_{-1}^1 \frac{x^3+|x|+1}{x^2+2|x|+1} dx$ is equal to

Let $\int_{-2}^{2} (\sin |x| + [x \sin x]) dx = 2(3 - \cos 2) + \beta$, where $[\cdot]$ denotes the greatest integer function. Then $\beta \sin \left(\frac{\beta}{2}\right)$ equals: