Antiderivative of function and general appearance. Lecture "An antiderivative. The concept of an antiderivative. The main property of an antiderivative function" (11th grade)

One of the operations of differentiation is finding the derivative (differential) and applying it to the study of functions.

The inverse problem is no less important. If the behavior of a function in the vicinity of each point of its definition is known, then how can one reconstruct the function as a whole, i.e. throughout the entire scope of its definition. This problem is the subject of study of the so-called integral calculus.

Integration is the inverse action of differentiation. Or restoring the function f(x) from a given derivative f`(x). The Latin word “integro” means restoration.

Example No. 1.

Let (f(x))’ = 3x 2. Let's find f(x).

Solution:

Based on the rule of differentiation, it is not difficult to guess that f(x) = x 3, because

(x 3)’ = 3x 2 However, it can be easily seen that f(x) is not found uniquely. As f(x), you can take f(x)= x 3 +1 f(x)= x 3 +2 f(x)= x 3 -3, etc.

Because the derivative of each of them is 3x 2. (The derivative of a constant is 0). All these functions differ from each other by a constant term. That's why general solution the problem can be written in the form f(x)= x 3 +C, where C is any constant real number.

Any of the found functions f(x) is called antiderivative for the function F`(x)= 3x 2

Definition.

A function F(x) is called antiderivative for a function f(x) on a given interval J if for all x from this interval F`(x)= f(x). So the function F(x)=x 3 is antiderivative for f(x)=3x 2 on (- ∞ ; ∞). Since for all x ~R the equality is true: F`(x)=(x 3)`=3x 2

As we have already noted, this function has infinite set primitives.

Example No. 2.

The function is antiderivative for all on the interval (0; +∞), because for all h from this interval, equality holds.

The task of integration is to find all its antiderivatives for a given function. When solving this problem, the following statement plays an important role:

A sign of constancy of function. If F"(x) = 0 on some interval I, then the function F is constant on this interval.

Proof.

Let us fix some x 0 from the interval I. Then for any number x from such an interval, by virtue of the Lagrange formula, we can indicate a number c contained between x and x 0 such that

F(x) - F(x 0) = F"(c)(x-x 0).

By condition, F’ (c) = 0, since c ∈1, therefore,

F(x) - F(x 0) = 0.

So, for all x from the interval I

that is, the function F maintains a constant value.

All antiderivative functions f can be written using one formula, which is called general form of antiderivatives for the function f. The following theorem is true ( main property of antiderivatives):

Theorem. Any antiderivative for a function f on the interval I can be written in the form

F(x) + C, (1) where F (x) is one of the antiderivatives for the function f (x) on the interval I, and C is an arbitrary constant.

Let us explain this statement, in which two properties of the antiderivative are briefly formulated:

Whatever number we put in expression (1) instead of C, we obtain the antiderivative for f on the interval I;
no matter what antiderivative Ф for f on the interval I is taken, it is possible to select a number C such that for all x from the interval I the equality

Proof.

By condition, the function F is antiderivative for f on the interval I. Therefore, F"(x)= f (x) for any x∈1, therefore (F(x) + C)" = F"(x) + C"= f(x)+0=f(x), i.e. F(x) + C is the antiderivative for the function f.
Let Ф (x) be one of the antiderivatives for the function f on the same interval I, i.e. Ф "(x) = f (х) for all x∈I.

Then (Ф(x) - F (x))" = Ф"(x)-F' (x) = f(x)-f(x)=0.

From here it follows c. the power of the sign of constancy of the function, that the difference Ф(х) - F(х) is a function that takes some constant value C on the interval I.

Thus, for all x from the interval I the equality Ф(x) - F(x)=С is true, which is what needed to be proved. The main property of the antiderivative can be given geometric meaning: graphs of any two antiderivatives for the function f are obtained from each other by parallel translation along the Oy axis

Questions for notes

The function F(x) is an antiderivative of the function f(x). Find F(1) if f(x)=9x2 - 6x + 1 and F(-1) = 2.

Find all antiderivatives for the function

For the function (x) = cos2 * sin2x, find the antiderivative of F(x) if F(0) = 0.

For a function, find an antiderivative whose graph passes through the point

Antiderivative

Definition of an antiderivative function

Function y=F(x) is called the antiderivative of the function y=f(x) at a given interval X, if for everyone X ∈X equality holds: F′(x) = f(x)

Can be read in two ways:

f derivative of a function F
F antiderivative of a function f

Property of antiderivatives

If F(x)- antiderivative of a function f(x) on a given interval, then the function f(x) has infinitely many antiderivatives, and all these antiderivatives can be written in the form F(x) + C, where C is an arbitrary constant.

Geometric interpretation

Graphs of all antiderivatives of a given function f(x) are obtained from the graph of any one antiderivative by parallel translations along the O axis at.

Rules for calculating antiderivatives

The antiderivative of the sum is equal to the sum of the antiderivatives. If F(x)- antiderivative for f(x), and G(x) is an antiderivative for g(x), That F(x) + G(x)- antiderivative for f(x) + g(x).
The constant factor can be taken out of the sign of the derivative. If F(x)- antiderivative for f(x), And k- constant, then k·F(x)- antiderivative for k f(x).
If F(x)- antiderivative for f(x), And k, b- constant, and k ≠ 0, That 1/k F(kx + b)- antiderivative for f(kx + b).

Remember!

Any function F(x) = x 2 + C , where C is an arbitrary constant, and only such a function is an antiderivative for the function f(x) = 2x.

For example:
F"(x) = (x 2 + 1)" = 2x = f(x);

f(x) = 2x, because F"(x) = (x 2 – 1)" = 2x = f(x);

f(x) = 2x, because F"(x) = (x 2 –3)" = 2x = f(x);

Relationship between the graphs of a function and its antiderivative:

If the graph of a function f(x)>0 F(x) increases over this interval.
If the graph of a function f(x)<0 on the interval, then the graph of its antiderivative F(x) decreases over this interval.
If f(x)=0, then the graph of its antiderivative F(x) at this point changes from increasing to decreasing (or vice versa).

To denote an antiderivative, use the sign indefinite integral, that is, an integral without indicating the limits of integration.

Indefinite integral

Definition:

The indefinite integral of the function f(x) is the expression F(x) + C, that is, the set of all antiderivatives of a given function f(x). The indefinite integral is denoted as follows: \int f(x) dx = F(x) + C

f(x)- called the integrand function;
f(x) dx- called the integrand;
x- called the variable of integration;
F(x)- one of the antiderivatives of the function f(x);
WITH- arbitrary constant.

Properties of the indefinite integral

The derivative of the indefinite integral is equal to the integrand: (\int f(x) dx)\prime= f(x) .
The constant factor of the integrand can be taken out of the integral sign: \int k \cdot f(x) dx = k \cdot \int f(x) dx.
The integral of the sum (difference) of functions is equal to the sum (difference) of the integrals of these functions: \int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx.
If k, b are constants, and k ≠ 0, then \int f(kx + b) dx = \frac(1)(k) \cdot F(kx + b) + C.

Table of antiderivatives and indefinite integrals

Function f(x)	Antiderivative F(x) + C	Indefinite integrals \int f(x) dx = F(x) + C
0	C	\int 0 dx = C
f(x) = k	F(x) = kx + C	\int kdx = kx + C
f(x) = x^m, m\not =-1	F(x) = \frac(x^(m+1))(m+1) + C	\int x(^m)dx = \frac(x^(m+1))(m+1) + C
f(x) = \frac(1)(x)	F(x) = l n \lvert x \rvert + C	\int \frac(dx)(x) = l n \lvert x \rvert + C
f(x) = e^x	F(x) = e^x + C	\int e(^x )dx = e^x + C
f(x) = a^x	F(x) = \frac(a^x)(l na) + C	\int a(^x )dx = \frac(a^x)(l na) + C
f(x) = \sin x	F(x) = -\cos x + C	\int \sin x dx = -\cos x + C
f(x) = \cos x	F(x) =\sin x + C	\int \cos x dx = \sin x + C
f(x) = \frac(1)(\sin (^2) x)	F(x) = -\ctg x + C	\int \frac (dx)(\sin (^2) x) = -\ctg x + C
f(x) = \frac(1)(\cos (^2) x)	F(x) = \tg x + C	\int \frac(dx)(\sin (^2) x) = \tg x + C
f(x) = \sqrt(x)	F(x) =\frac(2x \sqrt(x))(3) + C
f(x) =\frac(1)( \sqrt(x))	F(x) =2\sqrt(x) + C
f(x) =\frac(1)( \sqrt(1-x^2))	F(x)=\arcsin x + C	\int \frac(dx)( \sqrt(1-x^2))=\arcsin x + C
f(x) =\frac(1)( \sqrt(1+x^2))	F(x)=\arctg x + C	\int \frac(dx)( \sqrt(1+x^2))=\arctg x + C
f(x)=\frac(1)( \sqrt(a^2-x^2))	F(x)=\arcsin\frac (x)(a)+ C	\int \frac(dx)( \sqrt(a^2-x^2)) =\arcsin \frac (x)(a)+ C
f(x)=\frac(1)( \sqrt(a^2+x^2))	F(x)=\arctg \frac (x)(a)+ C	\int \frac(dx)( \sqrt(a^2+x^2)) = \frac (1)(a) \arctg \frac (x)(a)+ C
f(x) =\frac(1)( 1+x^2)	F(x)=\arctg + C	\int \frac(dx)( 1+x^2)=\arctg + C
f(x)=\frac(1)( \sqrt(x^2-a^2)) (a \not= 0)	F(x)=\frac(1)(2a)l n \lvert \frac (x-a)(x+a) \rvert + C	\int \frac(dx)( \sqrt(x^2-a^2))=\frac(1)(2a)l n \lvert \frac (x-a)(x+a) \rvert + C
f(x)=\tg x	F(x)= - l n \lvert \cos x \rvert + C	\int \tg x dx =- l n \lvert \cos x \rvert + C
f(x)=\ctg x	F(x)= l n \lvert \sin x \rvert + C	\int \ctg x dx = l n \lvert \sin x \rvert + C
f(x)=\frac(1)(\sin x)	F(x)= l n \lvert \tg \frac(x)(2) \rvert + C	\int \frac (dx)(\sin x) = l n \lvert \tg \frac(x)(2) \rvert + C
f(x)=\frac(1)(\cos x)	F(x)= l n \lvert \tg (\frac(x)(2) +\frac(\pi)(4)) \rvert + C	\int \frac (dx)(\cos x) = l n \lvert \tg (\frac(x)(2) +\frac(\pi)(4)) \rvert + C

Newton–Leibniz formula

Let f(x) this function F its arbitrary antiderivative.

\int_(a)^(b) f(x) dx =F(x)|_(a)^(b)= F(b) - F(a)

Where F(x)- antiderivative for f(x)

That is, the integral of the function f(x) on an interval is equal to the difference of antiderivatives at points b And a.

Area of a curved trapezoid

Curvilinear trapezoid is a figure bounded by the graph of a function that is non-negative and continuous on an interval f, Ox axis and straight lines x = a And x = b.

The area of a curved trapezoid is found using the Newton-Leibniz formula:

S= \int_(a)^(b) f(x) dx

Lesson and presentation on the topic: "An antiderivative function. Graph of a function"

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Antiderivative function. Introduction

Guys, you know how to find derivatives of functions using various formulas and rules. Today we will study the inverse operation of calculating the derivative. The concept of derivative is often used in real life. Let me remind you: the derivative is the rate of change of a function at a specific point. Processes involving motion and speed are well described in these terms.

Let's look at this problem: “The speed of an object moving in a straight line is described by the formula $V=gt$. It is required to restore the law of motion.
Solution.
We know the formula well: $S"=v(t)$, where S is the law of motion.
Our task comes down to finding a function $S=S(t)$ whose derivative is equal to $gt$. Looking carefully, you can guess that $S(t)=\frac(g*t^2)(2)$.
Let's check the correctness of the solution to this problem: $S"(t)=(\frac(g*t^2)(2))"=\frac(g)(2)*2t=g*t$.
Knowing the derivative of the function, we found the function itself, that is, we performed the inverse operation.
But it’s worth paying attention to this moment. The solution to our problem requires clarification; if we add any number (constant) to the found function, then the value of the derivative will not change: $S(t)=\frac(g*t^2)(2)+c,c=const$.
$S"(t)=(\frac(g*t^2)(2))"+c"=g*t+0=g*t$.

Guys, pay attention: our problem has an infinite number of solutions!
If the problem does not specify an initial or some other condition, do not forget to add a constant to the solution. For example, our task may specify the position of our body at the very beginning of the movement. Then it is not difficult to calculate the constant; by substituting zero into the resulting equation, we obtain the value of the constant.

What is this operation called?
The inverse operation of differentiation is called integration.
Finding a function from a given derivative – integration.
The function itself will be called an antiderivative, that is, the image from which the derivative of the function was obtained.
It is customary to write the antiderivative with a capital letter $y=F"(x)=f(x)$.

Definition. The function $y=F(x)$ is called the antiderivative of the function $у=f(x)$ on the interval X if for any $хϵХ$ the equality $F’(x)=f(x)$ holds.

Let's make a table of antiderivatives for various functions. It should be printed out as a reminder and memorized.

In our table, no initial conditions were specified. This means that a constant should be added to each expression on the right side of the table. We will clarify this rule later.

Rules for finding antiderivatives

Let's write down a few rules that will help us in finding antiderivatives. They are all similar to the rules of differentiation.

Rule 1. The antiderivative of a sum is equal to the sum of the antiderivatives. $F(x+y)=F(x)+F(y)$.

Example.
Find the antiderivative for the function $y=4x^3+cos(x)$.
Solution.
The antiderivative of the sum is equal to the sum of the antiderivatives, then we need to find the antiderivative for each of the presented functions.
$f(x)=4x^3$ => $F(x)=x^4$.
$f(x)=cos(x)$ => $F(x)=sin(x)$.
Then the antiderivative original function will be: $y=x^4+sin(x)$ or any function of the form $y=x^4+sin(x)+C$.

Rule 2. If $F(x)$ is an antiderivative for $f(x)$, then $k*F(x)$ is an antiderivative for the function $k*f(x)$.(We can easily take the coefficient as a function).

Example.
Find antiderivatives of functions:
a) $y=8sin(x)$.
b) $y=-\frac(2)(3)cos(x)$.
c) $y=(3x)^2+4x+5$.
Solution.
a) The antiderivative of $sin(x)$ is minus $cos(x)$. Then the antiderivative of the original function will take the form: $y=-8cos(x)$.

B) The antiderivative of $cos(x)$ is $sin(x)$. Then the antiderivative of the original function will take the form: $y=-\frac(2)(3)sin(x)$.

C) The antiderivative for $x^2$ is $\frac(x^3)(3)$. The antiderivative for x is $\frac(x^2)(2)$. The antiderivative of 1 is x. Then the antiderivative of the original function will take the form: $y=3*\frac(x^3)(3)+4*\frac(x^2)(2)+5*x=x^3+2x^2+5x$ .

Rule 3. If $у=F(x)$ is an antiderivative for the function $y=f(x)$, then the antiderivative for the function $y=f(kx+m)$ is the function $y=\frac(1)(k)* F(kx+m)$.

Example.
Find antiderivatives of the following functions:
a) $y=cos(7x)$.
b) $y=sin(\frac(x)(2))$.
c) $y=(-2x+3)^3$.
d) $y=e^(\frac(2x+1)(5))$.
Solution.
a) The antiderivative of $cos(x)$ is $sin(x)$. Then the antiderivative for the function $y=cos(7x)$ will be the function $y=\frac(1)(7)*sin(7x)=\frac(sin(7x))(7)$.

B) The antiderivative of $sin(x)$ is minus $cos(x)$. Then the antiderivative for the function $y=sin(\frac(x)(2))$ will be the function $y=-\frac(1)(\frac(1)(2))cos(\frac(x)(2) )=-2cos(\frac(x)(2))$.

C) The antiderivative for $x^3$ is $\frac(x^4)(4)$, then the antiderivative of the original function $y=-\frac(1)(2)*\frac(((-2x+3) )^4)(4)=-\frac(((-2x+3))^4)(8)$.

D) Slightly simplify the expression to the power $\frac(2x+1)(5)=\frac(2)(5)x+\frac(1)(5)$.
The antiderivative of an exponential function is the exponential function itself. The antiderivative of the original function will be $y=\frac(1)(\frac(2)(5))e^(\frac(2)(5)x+\frac(1)(5))=\frac(5)( 2)*e^(\frac(2x+1)(5))$.

Theorem. If $y=F(x)$ is an antiderivative for the function $y=f(x)$ on the interval X, then the function $y=f(x)$ has infinitely many antiderivatives, and all of them have the form $y=F( x)+С$.

If in all the examples considered above it was necessary to find the set of all antiderivatives, then the constant C should be added everywhere.
For the function $y=cos(7x)$ all antiderivatives have the form: $y=\frac(sin(7x))(7)+C$.
For the function $y=(-2x+3)^3$ all antiderivatives have the form: $y=-\frac(((-2x+3))^4)(8)+C$.

Example.
Given the law of change in the speed of a body over time $v=-3sin(4t)$, find the law of motion $S=S(t)$, if in starting moment time the body had a coordinate equal to 1.75.
Solution.
Since $v=S’(t)$, we need to find the antiderivative for a given speed.
$S=-3*\frac(1)(4)(-cos(4t))+C=\frac(3)(4)cos(4t)+C$.
In this problem it is given additional condition- initial moment of time. This means that $t=0$.
$S(0)=\frac(3)(4)cos(4*0)+C=\frac(7)(4)$.
$\frac(3)(4)cos(0)+C=\frac(7)(4)$.
$\frac(3)(4)*1+C=\frac(7)(4)$.
$C=1$.
Then the law of motion is described by the formula: $S=\frac(3)(4)cos(4t)+1$.

Problems to solve independently

1. Find antiderivatives of functions:
a) $y=-10sin(x)$.
b) $y=\frac(5)(6)cos(x)$.
c) $y=(4x)^5+(3x)^2+5x$.
2. Find antiderivatives of the following functions:
a) $y=cos(\frac(3)(4)x)$.
b) $y=sin(8x)$.
c) $y=((7x+4))^4$.
d) $y=e^(\frac(3x+1)(6))$.
3. According to the given law of change in the speed of a body over time $v=4cos(6t)$, find the law of motion $S=S(t)$ if at the initial moment of time the body had a coordinate equal to 2.

Antiderivative function and indefinite integral

Fact 1. Integration is the inverse action of differentiation, namely, restoring a function from the known derivative of this function. The function thus restored F(x) is called antiderivative for function f(x).

Definition 1. Function F(x f(x) on some interval X, if for all values x from this interval the equality holds F "(x)=f(x), that is, this function f(x) is the derivative of the antiderivative function F(x). .

For example, the function F(x) = sin x is an antiderivative of the function f(x) = cos x on the entire number line, since for any value of x (sin x)" = (cos x) .

Definition 2. Indefinite integral of a function f(x) is the set of all its antiderivatives. In this case, the notation is used

∫

f(x)dx

where is the sign ∫ called the integral sign, the function f(x) – integrand function, and f(x)dx – integrand expression.

Thus, if F(x) – some antiderivative for f(x) , That

∫

f(x)dx = F(x) +C

Where C - arbitrary constant (constant).

To understand the meaning of the set of antiderivatives of a function as an indefinite integral, the following analogy is appropriate. Let there be a door (traditional wooden door). Its function is to “be a door.” What is the door made of? Made of wood. This means that the set of antiderivatives of the integrand of the function “to be a door”, that is, its indefinite integral, is the function “to be a tree + C”, where C is a constant, which in this context can denote, for example, the type of tree. Just as a door is made from wood using some tools, a derivative of a function is “made” from an antiderivative function using formulas we learned while studying the derivative .

Then the table of functions of common objects and their corresponding antiderivatives (“to be a door” - “to be a tree”, “to be a spoon” - “to be metal”, etc.) is similar to the table of basic indefinite integrals, which will be given below. The table of indefinite integrals lists common functions, indicating the antiderivatives from which these functions are “made”. In part of the problems on finding the indefinite integral, integrands are given that can be integrated directly without much effort, that is, using the table of indefinite integrals. In more complex problems, the integrand must first be transformed so that table integrals can be used.

Fact 2. When restoring a function as an antiderivative, we must take into account an arbitrary constant (constant) C, and in order not to write a list of antiderivatives with various constants from 1 to infinity, you need to write a set of antiderivatives with an arbitrary constant C, for example, like this: 5 x³+C. So, an arbitrary constant (constant) is included in the expression of the antiderivative, since the antiderivative can be a function, for example, 5 x³+4 or 5 x³+3 and when differentiated, 4 or 3, or any other constant goes to zero.

Let us pose the integration problem: for this function f(x) find such a function F(x), whose derivative equal to f(x).

Example 1. Find the set of antiderivatives of a function

Solution. For this function, the antiderivative is the function

Function F(x) is called an antiderivative for the function f(x), if the derivative F(x) is equal to f(x), or, which is the same thing, differential F(x) is equal f(x) dx, i.e.

(2)

Therefore, the function is an antiderivative of the function. However, it is not the only antiderivative for . They also serve as functions

Where WITH– arbitrary constant. This can be verified by differentiation.

Thus, if there is one antiderivative for a function, then for it there is an infinite number of antiderivatives that differ by a constant term. All antiderivatives for a function are written in the above form. This follows from the following theorem.

Theorem (formal statement of fact 2). If F(x) – antiderivative for the function f(x) on some interval X, then any other antiderivative for f(x) on the same interval can be represented in the form F(x) + C, Where WITH– arbitrary constant.

In the next example, we turn to the table of integrals, which will be given in paragraph 3, after the properties of the indefinite integral. We do this before reading the entire table so that the essence of the above is clear. And after the table and properties, we will use them in their entirety during integration.

Example 2. Find sets of antiderivative functions:

Solution. We find sets of antiderivative functions from which these functions are “made”. When mentioning formulas from the table of integrals, for now just accept that there are such formulas there, and we will study the table of indefinite integrals itself a little further.

1) Applying formula (7) from the table of integrals for n= 3, we get

2) Using formula (10) from the table of integrals for n= 1/3, we have

3) Since

then according to formula (7) with n= -1/4 we find

It is not the function itself that is written under the integral sign f, and its product by the differential dx. This is done primarily in order to indicate by which variable the antiderivative is sought. For example,

, ;

here in both cases the integrand is equal to , but its indefinite integrals in the cases considered turn out to be different. In the first case, this function is considered as a function of the variable x, and in the second - as a function of z .

The process of finding the indefinite integral of a function is called integrating that function.

Geometric meaning of the indefinite integral

Suppose we need to find a curve y=F(x) and we already know that the tangent of the tangent angle at each of its points is a given function f(x) abscissa of this point.

According to the geometric meaning of the derivative, the tangent of the angle of inclination of the tangent at a given point of the curve y=F(x) equal to the value of the derivative F"(x). So we need to find such a function F(x), for which F"(x)=f(x). Function required in the task F(x) is an antiderivative of f(x). The conditions of the problem are satisfied not by one curve, but by a family of curves. y=F(x)- one of these curves, and any other curve can be obtained from it by parallel translation along the axis Oy.

Let's call the graph of the antiderivative function of f(x) integral curve. If F"(x)=f(x), then the graph of the function y=F(x) there is an integral curve.

Fact 3. The indefinite integral is geometrically represented by the family of all integral curves , as in the picture below. The distance of each curve from the origin of coordinates is determined by an arbitrary integration constant C.

Properties of the indefinite integral

Fact 4. Theorem 1. The derivative of an indefinite integral is equal to the integrand, and its differential is equal to the integrand.

Fact 5. Theorem 2. Indefinite integral of the differential of a function f(x) equal to function f(x) up to a constant term , i.e.

(3)

Theorems 1 and 2 show that differentiation and integration are mutually inverse operations.

Fact 6. Theorem 3. The constant factor in the integrand can be taken out of the sign of the indefinite integral , i.e.

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Antiderivative Indefinite integral

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Integration. Antiderivative. Continuous function F(x) calledantiderivativeForfunctions f (x) on the interval X if For each F’ (x) = f (x). EXAMPLE Function F(x) = x 3 is antiderivativeForfunctions f(x) = 3x...

SPECIAL EDUCATION OF THE USSR Approved by the Educational and Methodological Directorate for Higher Education HIGHER MATHEMATICS METHODICAL INSTRUCTIONS AND CONTROL TASKS (WITH THE PROGRAM) for part-time students of engineering and technical specialties

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Questions For self-test Define antiderivativefunctions. Indicate the geometric meaning of the aggregate primitivefunctions. What called uncertain...