Let $\int \frac{d x}{\sqrt{x+1}-\sqrt{x-1}}=\alpha(x+1)^{\frac{3}{2}}+$ $$ \beta(x-1)^{\frac{3}{2}}+c $$

What is the value of $\beta$?

Consider the following for the next two (02) items that follow:

The circle $x^2 + y^2 - 2x = 0$ is partitioned by line $y = x$ in two segments. Let $A_1, A_2$ be the areas of major and minor segments respectively.

What is the value of $A_1$?

The circle $x^2 + y^2 - 2x = 0$ is partitioned by line $y = x$ in two segments. Let $A_1, A_2$ be the areas of major and minor segments respectively.

What is the value of $\frac{2(A_1 + A_2)}{ A_1 - 3A_2}$?

Consider the following for the next two (02) items that follow :

Let $3f(x) + f\left(\frac{1}{x}\right) = \frac{1}{x} + 1$

What is $f(x)$ equal to?