What is "arbitrary constant" ? What is the meaning of 'arbitrary'?

Q. What is $\int (x\cos x+\sin x)dx$ equal to?

Q. What is the solution of the equation $ln\left(\frac{dy}{dx}\right)+x=0$? where c is an arbitrary constant.

Q. What is the number of arbitrary constants in the particular solution of differential equation of third order ?

Q. What is the solution of the differential equation $\frac{ydx-xdy}{y^2}=0$?
where $c$ is an arbitrary constant.

Q. What is the solution of the differential equation $\sin \left(\frac{dy}{dx}\right)-a=0$?
where $c$ is an arbitrary constant.