# How to take the definite integral on both sides of a differential equation?

For instance,

$$a \cdot ds=dt$$

I know that one can take the indefinite integral on both sides to get

$$\int a ds = \int 1 dt$$

But how do I take the definite integral of both sides, and exactly what do I need to know to do this? (Specifically, the bounds. How do I know what bounds to use?)

• You would do these as you would any other integrals. – 123 Jan 8 '15 at 3:08
• So, say if I know the bounds with respect to t will be 0 and t then the bounds of ds will be s(0) and s(t)? – Jason Jan 8 '15 at 3:10
• Indeed. As Saibal points out in his answer, you simply use initial conditions to establish bounds of integration. – 123 Jan 8 '15 at 3:11

Suppose you have the initial condition $t=t_0\implies s=s_0$. Then you can integrate: $$\int_{s_0}^{s}ads = \int_{t_0}^{t}dt$$ This is equivalent to first evaluating the indefinite integral and then solving for the constant of integration.