$\int_0^2 |1-x^2| dx = $

$\int_{0}^{1} \frac{x^{2}}{1+x^{2}} \, dx =$

The value of $\int_3^5 {\frac{{{x^2}}}{{{x^2} - 4}}\,dx} $ is

If $p(x)$ is a cubic polynomial with $p(1)=3, p(0)=2$ and $p(-1)=4$,then $\int_{-1}^1 p(x) dx$ is

If $\int_0^{\frac{\pi}{2}} \frac{\cot x}{\cot x + \csc x} dx = m(\pi + n)$,then $m \cdot n$ is equal to

Let $\alpha \in (0,1)$ and $\beta = \log_{e}(1-\alpha)$. Let $P_n(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \dots + \frac{x^n}{n}$ for $x \in (0,1)$. Then the integral $\int_{0}^{\alpha} \frac{t^{50}}{1-t} dt$ is equal to