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

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

What is u + v equal to?

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?