For the following two (02) items:
Let $u=\int e^{x}cos~x~dx $ and $v=\int e^{x}sin~x~dx.$

Consider the following:
I. $\frac{du}{dx}=-v $
II. $\frac{dv}{dx}=-u$
Which of the above is/are correct?

 What is $\int \frac{dx}{10^{x}+10^{-x}}$ equal to?

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

Let the function y = (1 - cos x)^-1 where$x \ne 2n\pi$ and n is an integer

What is ∫ydx equal to? 

where c is the constant of integration.

Direction : Consider the following for the items that follow :

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

What is |U²(x) - V²(x)| equal to?