Reversing the Effects of Differentiation

Integration is used frequently in kinetics, thermodynamics, quantum mechanics and other areas of chemistry, where we build models based on changing quantities. Thus, if we know the rate of change of a property, y (the dependent variable), with respect to x (the independent variable), in the form of dy/dx, then integral calculus provides us with the tools for obtaining the form of y as a function of x. We see that integration reverses the effects of differentiation. 

Consider, for example, a car undergoing a journey with an initial speed u and moving with a constant acceleration a. The speed, v, and distance, s, travelled after time t are given by  

The rate of change of distance with time yields the speed, v at time, t  

However, the reverse process, in going from speed to distance, involves integration of the rate equation (6.2). In chemistry, the concept of rate is central to an understanding of chemical kinetics, in which we have to deal with analogous rate equations which typically involve the rate of change of concentration, rather than the rate of change of distance. For example, in a first-order chemical reaction, where the rate of loss of the reactant is proportional to the concentration of the reactant, the rate equation takes the form  

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