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, the sign of an indefinite integral is used, 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.
- Integral of the sum (difference) of functions equal to the sum(differences) of 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
Solving integrals is an easy task, but only for a select few. This article is for those who want to learn to understand integrals, but know nothing or almost nothing about them. Integral... Why is it needed? How to calculate it? What is certain and indefinite integral s? If the only use you know of for an integral is to use a crochet hook shaped like an integral icon to get something useful out of hard-to-reach places, then welcome! Find out how to solve integrals and why you can't do without it.
We study the concept of "integral"
Integration was known back in Ancient Egypt. Of course, not in its modern form, but still. Since then, mathematicians have written many books on this topic. Especially distinguished themselves Newton And Leibniz , but the essence of things has not changed. How to understand integrals from scratch? No way! To understand this topic, you will still need a basic knowledge of the basics of mathematical analysis. It is this fundamental information that you will find on our blog.
Indefinite integral
Let us have some function f(x) .
Indefinite integral function f(x) this function is called F(x) , whose derivative is equal to the function f(x) .
In other words, an integral is a derivative in reverse or an antiderivative. By the way, read about how in our article.
An antiderivative exists for all continuous functions. Also, a constant sign is often added to the antiderivative, since the derivatives of functions that differ by a constant coincide. The process of finding the integral is called integration.
Simple example:
In order not to constantly calculate antiderivatives of elementary functions, it is convenient to put them in a table and use ready-made values:
Definite integral
When dealing with the concept of an integral, we are dealing with infinitesimal quantities. The integral will help to calculate the area of a figure, the mass of a non-uniform body, the distance traveled during uneven movement, and much more. It should be remembered that an integral is the sum of an infinitely large number of infinitesimal terms.
As an example, imagine a graph of some function. How to find the area of a figure bounded by the graph of a function?
Using an integral! Let us divide the curvilinear trapezoid, limited by the coordinate axes and the graph of the function, into infinitesimal segments. This way the figure will be divided into thin columns. The sum of the areas of the columns will be the area of the trapezoid. But remember that such a calculation will give an approximate result. However, the smaller and narrower the segments, the more accurate the calculation will be. If we reduce them to such an extent that the length tends to zero, then the sum of the areas of the segments will tend to the area of the figure. This is a definite integral, which is written like this:
Points a and b are called limits of integration.
Bari Alibasov and the group "Integral"
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Rules for calculating integrals for dummies
Properties of the indefinite integral
How to solve an indefinite integral? Here we will look at the properties of the indefinite integral, which will be useful in solving examples.
- The derivative of the integral is equal to the integrand:
- The constant can be taken out from under the integral sign:
- The integral of the sum is equal to the sum of the integrals. This is also true for the difference:
Properties of a definite integral
- Linearity:
- The sign of the integral changes if the limits of integration are swapped:
- At any points a, b And With:
We have already found out that a definite integral is the limit of a sum. But how to get a specific value when solving an example? For this there is the Newton-Leibniz formula:
Examples of solving integrals
Below we will consider several examples of finding indefinite integrals. We invite you to figure out the intricacies of the solution yourself, and if something is unclear, ask questions in the comments.
To reinforce the material, watch a video about how integrals are solved in practice. Don't despair if the integral is not given right away. Ask and they will tell you everything they know about calculating integrals. With our help, any triple or curvilinear integral over a closed surface will be within your power.
There are three basic rules for finding antiderivative functions. They are very similar to the corresponding differentiation rules.
Rule 1
If F is an antiderivative for some function f, and G is an antiderivative for some function g, then F + G will be an antiderivative for f + g.
By definition of an antiderivative, F’ = f. G' = g. And since these conditions are met, then according to the rule for calculating the derivative for the sum of functions we will have:
(F + G)’ = F’ + G’ = f + g.
Rule 2
If F is an antiderivative for some function f, and k is some constant. Then k*F is the antiderivative of the function k*f. This rule follows from the rule for calculating the derivative of a complex function.
We have: (k*F)’ = k*F’ = k*f.
Rule 3
If F(x) is some antiderivative for the function f(x), and k and b are some constants, and k is not equal to zero, then (1/k)*F*(k*x+b) will be an antiderivative for the function f (k*x+b).
This rule follows from the rule for calculating the derivative of a complex function:
((1/k)*F*(k*x+b))’ = (1/k)*F’(k*x+b)*k = f(k*x+b).
Let's look at a few examples of how these rules apply:
Example 1. Find the general form of antiderivatives for the function f(x) = x^3 +1/x^2. For the function x^3 one of the antiderivatives will be the function (x^4)/4, and for the function 1/x^2 one of the antiderivatives will be the function -1/x. Using the first rule, we have:
F(x) = x^4/4 - 1/x +C.
Example 2. Let's find the general form of antiderivatives for the function f(x) = 5*cos(x). For the function cos(x), one of the antiderivatives will be the function sin(x). If we now use the second rule, we will have:
F(x) = 5*sin(x).
Example 3. Find one of the antiderivatives for the function y = sin(3*x-2). For the function sin(x) one of the antiderivatives will be the function -cos(x). If we now use the third rule, we obtain an expression for the antiderivative:
F(x) = (-1/3)*cos(3*x-2)
Example 4. Find the antiderivative for the function f(x) = 1/(7-3*x)^5
The antiderivative for the function 1/x^5 will be the function (-1/(4*x^4)). Now, using the third rule, we get.
We have seen that the derivative has numerous uses: the derivative is the speed of movement (or, more generally, the speed of any process); derivative is the slope of the tangent to the graph of the function; using the derivative, you can examine the function for monotonicity and extrema; the derivative helps solve optimization problems.
But in real life we also have to solve inverse problems: for example, along with the problem of finding the speed according to a known law of motion, we also encounter the problem of restoring the law of motion according to a known speed. Let's consider one of these problems.
Example 1. A material point moves in a straight line, its speed at time t is given by the formula u = tg. Find the law of motion.
Solution. Let s = s(t) be the desired law of motion. It is known that s"(t) = u"(t). This means that to solve the problem you need to choose function s = s(t), whose derivative is equal to tg. It's not hard to guess that
Let us immediately note that the example is solved correctly, but incompletely. We found that, in fact, the problem has infinitely many solutions: any function of the form 
an arbitrary constant can serve as a law of motion, since
To make the task more specific, we needed to fix the initial situation: indicate the coordinate of a moving point at some point in time, for example, at t=0. If, say, s(0) = s 0, then from the equality we obtain s(0) = 0 + C, i.e. S 0 = C. Now the law of motion is uniquely defined:
In mathematics, mutually inverse operations are given different names and special notations are invented: for example, squaring (x 2) and taking the square root of sine (sinх) and arcsine(arcsin x), etc. The process of finding the derivative with respect to a given function is called differentiation, and the inverse operation, i.e. the process of finding a function from a given derivative - integration.
The term “derivative” itself can be justified “in everyday life”: the function y - f(x) “gives birth” to a new function y"= f"(x). The function y = f(x) acts as a “parent” , but mathematicians, naturally, do not call it a “parent” or “producer”; they say that this, in relation to the function y"=f"(x), is the primary image, or, in short, the antiderivative.
Definition 1. The function y = F(x) is called antiderivative for the function y = f(x) on a given interval X if for all x from X the equality F"(x)=f(x) holds.
In practice, the interval X is usually not specified, but is implied (as the natural domain of definition of the function).
Here are some examples:
1) The function y = x 2 is antiderivative for the function y = 2x, since for all x the equality (x 2)" = 2x is true.
2) the function y - x 3 is antiderivative for the function y-3x 2, since for all x the equality (x 3)" = 3x 2 is true.
3) The function y-sinх is an antiderivative for the function y = cosx, since for all x the equality (sinx)" = cosx is true.
4) The function is antiderivative for a function on the interval since for all x > 0 the equality is true
In general, knowing the formulas for finding derivatives, it is not difficult to compile a table of formulas for finding antiderivatives.
We hope you understand how this table is compiled: the derivative of the function that is written in the second column is equal to the function that is written in the corresponding row of the first column (check it, don’t be lazy, it’s very useful). For example, for the function y = x 5 the antiderivative, as you will establish, is the function (see the fourth row of the table).
Notes: 1. Below we will prove the theorem that if y = F(x) is an antiderivative for the function y = f(x), then the function y = f(x) has infinitely many antiderivatives and they all have the form y = F(x ) + C. Therefore, it would be more correct to add the term C everywhere in the second column of the table, where C is an arbitrary real number.
2. For the sake of brevity, sometimes instead of the phrase “the function y = F(x) is an antiderivative of the function y = f(x),” they say F(x) is an antiderivative of f(x).”
2. Rules for finding antiderivatives
When finding antiderivatives, as well as when finding derivatives, not only formulas are used (they are listed in the table on p. 196), but also some rules. They are directly related to the corresponding rules for calculating derivatives.
We know that the derivative of a sum is equal to the sum of its derivatives. This rule generates the corresponding rule for finding antiderivatives.
Rule 1. The antiderivative of a sum is equal to the sum of the antiderivatives.
We draw your attention to the somewhat “lightness” of this formulation. In fact, one should formulate the theorem: if the functions y = f(x) and y = g(x) have antiderivatives on the interval X, respectively y-F(x) and y-G(x), then the sum of the functions y = f(x)+g(x) has an antiderivative on the interval X, and this antiderivative is the function y = F(x)+G(x). But usually, when formulating rules (and not theorems), they leave only keywords- this makes it more convenient to apply the rule in practice
Example 2. Find the antiderivative for the function y = 2x + cos x.
Solution. The antiderivative for 2x is x"; the antiderivative for cox is sin x. This means that the antiderivative for the function y = 2x + cos x will be the function y = x 2 + sin x (and in general any function of the form Y = x 1 + sinx + C) .
We know that the constant factor can be taken out of the sign of the derivative. This rule generates the corresponding rule for finding antiderivatives.
Rule 2. The constant factor can be taken out of the sign of the antiderivative.
Example 3.
Solution. a) The antiderivative for sin x is -soz x; This means that for the function y = 5 sin x the antiderivative function will be the function y = -5 cos x.
b) The antiderivative for cos x is sin x; This means that the antiderivative of a function is the function
c) The antiderivative for x 3 is the antiderivative for x, the antiderivative for the function y = 1 is the function y = x. Using the first and second rules for finding antiderivatives, we find that the antiderivative for the function y = 12x 3 + 8x-1 is the function
Comment. As is known, the derivative of a product is not equal to the product of derivatives (the rule for differentiating a product is more complex) and the derivative of a quotient is not equal to the quotient of derivatives. Therefore, there are no rules for finding the antiderivative of the product or the antiderivative of the quotient of two functions. Be careful!
Let us obtain another rule for finding antiderivatives. We know that the derivative of the function y = f(kx+m) is calculated by the formula
This rule generates the corresponding rule for finding antiderivatives.
Rule 3. If y = F(x) is an antiderivative for the function y = f(x), then the antiderivative for the function y=f(kx+m) is the function
Indeed,
This means that it is an antiderivative for the function y = f(kx+m).
The meaning of the third rule is as follows. If you know that the antiderivative of the function y = f(x) is the function y = F(x), and you need to find the antiderivative of the function y = f(kx+m), then proceed like this: take the same function F, but instead of the argument x, substitute the expression kx+m; in addition, do not forget to write “correction factor” before the function sign
Example 4. Find antiderivatives for given functions:
Solution, a) The antiderivative for sin x is -soz x; This means that for the function y = sin2x the antiderivative will be the function
b) The antiderivative for cos x is sin x; This means that the antiderivative of a function is the function
c) The antiderivative for x 7 means that for the function y = (4-5x) 7 the antiderivative will be the function
3. Indefinite integral
We have already noted above that the problem of finding an antiderivative for a given function y = f(x) has more than one solution. Let's discuss this issue in more detail.
Proof. 1. Let y = F(x) be the antiderivative for the function y = f(x) on the interval X. This means that for all x from X the equality x"(x) = f(x) holds. Let us find the derivative of any function of the form y = F(x)+C:
(F(x) +C) = F"(x) +C = f(x) +0 = f(x).
So, (F(x)+C) = f(x). This means that y = F(x) + C is an antiderivative for the function y = f(x).
Thus, we have proven that if the function y = f(x) has an antiderivative y=F(x), then the function (f = f(x) has infinitely many antiderivatives, for example, any function of the form y = F(x) +C is an antiderivative.
2. Let us now prove that the indicated type of functions exhausts the entire set of antiderivatives.
Let y=F 1 (x) and y=F(x) be two antiderivatives for the function Y = f(x) on the interval X. This means that for all x from the interval X the following relations hold: F^ (x) = f (X); F"(x) = f(x).
Let's consider the function y = F 1 (x) -.F(x) and find its derivative: (F, (x) -F(x))" = F[(x)-F(x) = f(x) - f(x) = 0.
It is known that if the derivative of a function on an interval X is identically equal to zero, then the function is constant on the interval X (see Theorem 3 from § 35). This means that F 1 (x) - F (x) = C, i.e. Fx) = F(x)+C.
The theorem has been proven.
Example 5. The law of change of speed over time is given: v = -5sin2t. Find the law of motion s = s(t), if it is known that at time t=0 the coordinate of the point was equal to the number 1.5 (i.e. s(t) = 1.5).
Solution. Since speed is a derivative of the coordinate as a function of time, we first need to find the antiderivative of the speed, i.e. antiderivative for the function v = -5sin2t. One of such antiderivatives is the function , and the set of all antiderivatives has the form:
To find the specific value of the constant C, we use the initial conditions, according to which s(0) = 1.5. Substituting the values t=0, S = 1.5 into formula (1), we get:
Substituting the found value of C into formula (1), we obtain the law of motion that interests us:
Definition 2. If a function y = f(x) has an antiderivative y = F(x) on an interval X, then the set of all antiderivatives, i.e. the set of functions of the form y = F(x) + C is called the indefinite integral of the function y = f(x) and is denoted by:
(read: “indefinite integral ef from x de x”).
In the next paragraph we will find out what is hidden meaning the indicated designation.
Based on the table of antiderivatives available in this section, we will compile a table of the main indefinite integrals:
Based on the above three rules for finding antiderivatives, we can formulate the corresponding rules of integration.
Rule 1. The integral of the sum of functions is equal to the sum of the integrals of these functions:
Rule 2. The constant factor can be taken out of the integral sign:
Rule 3. If
Example 6. Find indefinite integrals:
Solution, a) Using the first and second rules of integration, we obtain:
Now let's use the 3rd and 4th integration formulas:
As a result we get:
b) Using the third rule of integration and formula 8, we obtain:
c) To directly find a given integral, we have neither the corresponding formula nor the corresponding rule. In such cases, previously performed identical transformations of the expression contained under the integral sign sometimes help.
Let's take advantage trigonometric formula Degree reduction:
Then we find sequentially:
A.G. Mordkovich Algebra 10th grade
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