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

If $[\cdot]$ denotes the greatest integer function,then the integral $\int_{0}^{\pi} [\cos x] \, dx$ is equal to:

The value of $\int_{0}^{\pi / 2} \frac{\sin^3 x}{\sin x + \cos x} dx$ is

The value of the integral $\int_{-1 / 2}^{1 / 2}\left\{\left(\frac{x+1}{x-1}\right)^{2}+\left(\frac{x-1}{x+1}\right)^{2}-2\right\}^{1 / 2} d x$ is equal to

Value of the definite integral $\int_{-3}^{1} (2(t+1)^5 - 5(t+1)^3 + t + 3) dt$ is equal to

If ${I_n} = \int_0^{\pi /4} {{\tan ^n}\theta \,d\theta ,} $ then for any positive integer $n,$ the value of $n({I_{n - 1}} + {I_{n + 1}})$ is