What is $\int _{n}^{n+1}(x-[x])dx$, where [.] is the greatest integer function and n is natural number?

Consider the following statements:

I. $y=xe^{2x}$ is the solution of $\frac{dy}{dx}=y(2+\frac{1}{x})$

II. $y=x\ln |x|+cx$ is the solution of $\frac{dy}{dx}=\frac{x+y}{x}$

Which of the statements given above is/are correct?

If k is an arbitrary constant, then what is the general solution of the equation $(x+y)^{2}\frac{dy}{dx}=k^{2}$

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