Antidifferentiation is also known as indefinite integration. It let us find a kind of function that is another kind of function’s antiderivative. In another word, antidifferentiation is the inverse of differential that produces a formula. For example, let [F(X)+C]’=f(x), the function [F(X)+C] is a series of functions that represents antidifferentiation of the function f(x). We can find a lot of constants C to get different functions of F(X)+C. These functions we get will have the same slope, but the values of x from the various function will be different.
The example of anti-differentiation: let f(x)=x, then the anti-differentiation of f(x) will be F(x)= +c. (The graph shows examples of the functions F(X)+C.)
Integration is a way to calculate the sum. In integration, we calculate the total area under the curve by separating the pattern under the curve into many rectangles. When we calculate the area under the curve, we will need the interval of x which is the total width of rectangles. We can finally get a value of a function by integration
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As we can see from the graph above, if we want to calculate the sum area under the curve between a and b, we can find the antiderivative of f(x) firstly, we suppose the antiderivative function is F(x)+C, and then the total area will be F(a)+C-[F(b)+C], and then, we can get an exact value.
Antidifferentiation is different from integration. As we can see, we can get a particular value by integration, and we can get a series of functions by antidifferentiation. At the same time, anti-differentiation and integration are related. We need to use the function we get through anti-differentiation to calculate integration. The constant C will be canceled when we do the calculation. Therefore, we can also consider that antidifferentiation is a step of integration.
The similar ways to change the original function that might cause students commonly mistake one from the other. Both anti-differentiation and integration make people change the original function into another function or particular value in which are related to each other.