The value of $\int_0^{1/2} \frac{dx}{\sqrt{1-x^{2n}}}$ is $(n \in N)$

$\int_0^2 \frac{x^3}{(x^2 + 1)^{3/2}} \, dx = $

Evaluate the definite integral: $\int_0^{\pi^2 / 4} (2 \sin \sqrt{x} + \sqrt{x} \cos \sqrt{x}) \, dx$

The value of $\int_{0}^{1} \frac{2x^2 + 3x + 3}{(x + 1)(x^2 + 2x + 2)} dx$ is:

For a real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$. The smallest positive integer $n$ for which the integral $\int_{1}^{n} [x][\sqrt{x}] \, dx$ exceeds $60$ is

Let $f : [1/2, 1] \rightarrow \mathbb{R}$ be a positive,non-constant and differentiable function such that $f^{\prime}(x) < 2f(x)$ and $f(1/2) = 1$. Then the value of $\int_{1/2}^1 f(x) dx$ lies in the interval