Why a constant is written after integrating?
Integration is the opposite of differentiation (taking the
derivative). The derivative of a constant is zero. Integration is
also called antidifferentiation since integration and
differentiation are opposites of each other. The derivative of x^2
is 2x. The antiderivative (integral) of 2x is x^2. However, the
derivative of x^2 + 7 is also 2x.
Therefore, the antiderivative of 2x is x^2 + C, in general,
where the constant C has to be determined from the context of the
problem. In the above case, the constant happens to be C=7. We use
integration to solve first order differential equations. When
solving first order differential equations, like in "word
problems", you must determine the integration constant using the
initial conditions (ie the conditions we know to be true at t=0 -
we usually know what these are), or the boundary conditions (ie the
conditions we know to be true at x=0 and y=0).