Integration is the inverse operation of differentiation, that is, knowing the derivative function of a function and finding the original function in reverse. Integral is widely used in summation, that is, to find the area of a curved triangle. This ingenious solution is determined by the special properties of integral. The indefinite integral of a function (also known as the original function) refers to another family of functions, and the derivative function of this family of functions is just the previous function.
The derivation process of differential, integral and derivative of extended data;
Let the function y = f(x) be defined in the neighborhood of X, and both X and x+δ x are in this interval. If the increment of the function Δ y = f (x+Δ x)-f (x) can be expressed as Δ y = a Δ x+o (Δ x) (where a is a constant that does not change with Δ x, but a can change with x), o (Δ x) is infinitely smaller than the higher order of Δ x.
Then the function f(x) is differentiable at point X, and aδX is called the differential of the function corresponding to the dependent variable increment δy at point X, which is denoted as dy, that is, dy = aδX X x. The differential of the function is the main part of the function increment and is a linear function of δ x, so the differential of the function is the linear main part of the function increment (δ x→ 0).
Let the function y = f(x) be defined in a certain interval, and x0 and x0+△x are in this interval. If the increment of the function Δ y = f (x0+△ x)? F(x0) can be expressed as Δ y = a Δ x+o (Δ x), where a is a constant independent of Δ x and o (Δ x) is an infinitesimal higher order of Δ x, then the function y = f o(δx) is differentiable at point x0. A Δ x is called the differential of the function at x0, which corresponds to the increment of the independent variable Δ x.
References:
Baidu Encyclopedia-Difference
Baidu encyclopedia-integral
Baidu encyclopedia-derivative