Integrals and Equations Involving Step Functions

Hello Everyone!

I've recently came across step functions in my systems course, they frequently appear in ODEs describing the system.

An ODE I came across was $\displaystyle y'+y=u(x)$. Now, I know I must ask this in the differential equations forum, but, I know what the solution should be, it's just that I'm not "okay" with integrals involving step functions. In the above equation, we would get $\displaystyle y = \displaystyle \frac{\int e^x . u(x)dx}{e^x}$ but what's $\displaystyle \displaystyle \int e^x .u(x)dx$? Shouldn't it be plain $\displaystyle u(x) . dx$?

If so, when substituting our new solution in the DE, we get: $\displaystyle \delta (x) + u(x) \neq u(x)$. How come?!

Any help is appreciated!