What is the original function? What does the original function mean?

Definition of the original function.

primitive function The known function f(x) is a function defined in an interval where there is a function f(x) such that there is any point in the interval.

df(x)=f(x)dx, then the function f(x) is called the original function of the function f(x) in this interval.

Example: sinx is the original function of cosx.

Questions about primitive functions.

What are the conditions for the function f(x) to ensure that its original function must exist? We'll solve this problem later. If there are primitive functions, how many primitive functions are there?

We can clearly see that if the function f(x) is the original function of the function f(x), i.e., f'(x)=f(x), then any function in the function family f(x)+c (c is any constant) must be the original function of f(x), so:

If the function f(x) has a primitive function, then its primitive function is infinitely plentiful.

If the functions f(x) and f(x) defined on (a,b) satisfy the conditions: for each x (a,b), f(x) f(x)?f(x) is called a primitive function of f(x).

For example, x3 is a primitive function of 3x2, and it is easy to know that x3 1 and x3 2 are also primitive functions of 3x2. Therefore, if a function has a primitive function, there are many primitive functions, and the concept of primitive function is proposed to solve the inverse operation of derivation and differentiation, for example: it is known that the velocity of an object moving in a straight line at any time t is v v(t), and its motion law is required, that is, the original function of v v(t) is found.

The problem of the existence of the original function is a fundamental theoretical problem in calculus, and when f(x) is a continuous function, its original function must exist.

Concepts in calculus. In layman's terms, if f(x) is a derivative function of f(x), then f(x) is the original function of f(x), and the original function can be obtained from an indefinite integral.

f(x) is the derivative of the original function f(x), and f(x)dx is the differentiation of the original function f(x), because d[f(x)].

For example, x3 is a primitive function of 3x2, and it is easy to know that x3 1 and x3 2 are also primitive functions of 3x2. Therefore, if a function has one original function, there are many many original functions, and the concept of the original function is proposed to solve the inverse operation of derivative and differentiation.

For example, if it is known that the velocity of an object moving in a straight line at any time t is v=v(t), the law of its motion is required to find the original function of v=v(t). The problem of the existence of the original function is the basic theoretical problem of calculus, when f(x) is a continuous function.

, its original function must exist.

Proto-function existence theorem:

Let the domain of the function f(x) be defined.

for d. If there is a positive sail t such that for any of them there is, and f(x+t)=f(x) is constant, then f(x) is said to be a periodic function.

t is called the period of f(x), and usually we say that the period of the periodic function refers to the minimum positive period.

The domain d of the periodic function is an unbounded interval on at least one side, and if d is bounded, the function is not periodic. Not every periodic function has a minimum positive period, such as the Dirichlet function.

1. Continuous functions must have original functions.

Second, when the function is discontinuous, it is known from Dab's theorem that if there is an original function in a discontinuous function, then the discontinuity point of this function is not a discontinuity point, the second is not a jump discontinuity point, and the third is not an infinite discontinuity point.

3. Discontinuous functions with ** discontinuity points do not necessarily exist as original functions, such as the piecewise tremor function.

f(x)=(1 x)*(sin1 x), (when x is not equal to 0); f(x)=0, (when x=0).The piecewise function f(x) has a break point x=0, but f(x) does not have the original function on any of the intervals [a,b] containing x=0 points.

The general textbook mentions the original function in two places:

1.Primitive functions and indefinite integrals.

Definition of the original function: if there is f (x) = f (x) for any point x on the interval i, then the function f(z) is said to be an original function of the function f(x) on the area infiltration mu i ;

2.Image of the original letter plexus macrosen number and its inverse function.

The relationship between y=x symmetries is eliminated.

The details are as follows:

For a known function f(x) defined in an interval, if there is a derivative function f(x), and the finch is such that df(x)=f(x)dx exists at any point in the interval, then the function f(x) is said to be the original function of the function f(x) in that interval.

Formula for indefinite integrals:

1. adx=ax+c, a and c are constants.

2. x adx=[x (a+1)] a+1)+c, where a is a constant and a≠-1

3、∫1/xdx=ln|x|+c

4. A xdx = (1 lna) a x+c, where a >0 and a ≠15, e xdx = e x+c

6、∫cosxdx=sinx+c

7、∫sinxdx=-cosx+c

8、∫cotxdx=ln|sinx|+c=-ln|cscx|+c

Find the integral for (1+x 2).

Make a triangular substitution, so that x=tant

then (1+x)dx

secttant+ln sect+tant --sect) 3dt, so (sect) 3dx=1 2(secttant+ln sect+tant) +c

Thus (1 hare chang x 2) dx

1/2（x√（1+x²）+ln（x+√（1+x²））c<>