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 . 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 but what's ? Shouldn't it be plain ?

If so, when substituting our new solution in the DE, we get: . How come?!

Any help is appreciated!