The value of the integral $\int_0^1 e^{x^2} dx$ lies in the interval

Let $[x]$ denote the greatest integer $\leq x$. Consider the function $f(x) = \max \{x^2, 1 + [x]\}$. Then the value of the integral $\int_0^2 f(x) dx$ is:

$\int\limits_0^{\frac{\pi }{2}} \frac{dx}{1 + a^2 \sin^2 x}$ has the value :

The value of $\int_0^1 4 x^3 \left\{ \frac{d^2}{d x^2} (1-x^2)^5 \right\} d x$ is

$\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{d x}{1+\cos x}$ is equal to

If $I_n = \int_0^{\frac{\pi}{2}} \cos^n x \cos(nx) dx$, then $I_1, I_2, I_3, \ldots$ are in