Evaluate the definite integral $\int_{-1}^{2} \left[ \frac{[x]}{1 + x^2} \right] dx$,where $[\cdot]$ denotes the Greatest Integer Function $(GIF)$.

Evaluate the definite integral: $\int_0^{\pi /8} \frac{\sec^2 2x}{2} \, dx$

The value of the integral $\int_{0}^{\pi / 4} \frac{\sin x+\cos x}{3+\sin 2 x} d x$ is equal to

The value of the integral $\int_{\pi/6}^{\pi/3} \left(\frac{4 - \csc^2 x}{\cos^4 x}\right) dx$ is:

Let $f(x) = \int_0^x t(t^2 - 9t + 20) dt$,$1 \leq x \leq 5$. If the range of $f$ is $[\alpha, \beta]$,then $4(\alpha + \beta)$ equals:

If the integral $525 \int_0^{\frac{\pi}{2}} \sin 2 x \cos^{\frac{11}{2}} x \left(1+\cos^{\frac{5}{2}} x\right)^{\frac{1}{2}} d x$ is equal to $(n \sqrt{2}-64)$,then $n$ is equal to