Indefinite Integration — Class 12 Mathematics PYQs

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

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

What is the value of $\beta$?

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

Let
$$2\int \frac{x^2-1}{\sqrt{x^2+1}}dx=U(x)V(x)-3\ln |U(x)+V(x)|+c$$

What is $|U^2(x)-V^2(x)|$ equal to?

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

Let
$$2\int \frac{x^2-1}{\sqrt{x^2+1}}dx=U(x)V(x)-3\ln |U(x)+V(x)|+c$$

What is $U(x)V(x)$ equal to?

What is $\int (\sin x{)}^{-1/2}(\cos x{)}^{-3/2}dx$ equal to ?

If $I_1=\int \frac{e^{x}dx}{e^x+e^{-x}}$ and $I_2=\int \frac{dx}{e^{2x}+1}$,
then what is $I_1+I_2$ equal to ?

What is

$$\int \frac{dx}{\sec x+\tan x}$$

equal to?

What is

$$\int \frac{dx}{{\sec }^2({\tan }^{-1}x)}$$

equal to?

What is $\int e^{(2\ln x+\ln x^2)}dx$ equal to?