Table of all integrals. Basic integration methods

Table of antiderivatives ("integrals"). Table of integrals. Tabular indefinite integrals. (The simplest integrals and integrals with a parameter). Formulas for integration by parts. Newton-Leibniz formula.

Formulas for integration by parts. Integration rules.

Table of derivatives. Tabular derivatives. Derivative of the product. Derivative of the quotient. Derivative of a complex function.

If x is an independent variable, then:

Properties of logarithms. Basic formulas for logarithms. Decimal (lg) and natural logarithms (ln).

Natural logarithm ln (logarithm to base e = 2.718281828459045...) ln(e)=1; ln(1)=0

Taylor series. Taylor series expansion of a function.

It turns out that the majority practically encountered mathematical functions can be represented with any accuracy in the vicinity of a certain point in the form of power series containing powers of a variable in increasing order. For example, in the vicinity of the point x=1:

When using series called Taylor's rows, mixed functions containing, say, algebraic, trigonometric and exponential functions can be expressed as purely algebraic functions. Using series, you can often quickly perform differentiation and integration.

The Taylor series in the neighborhood of point a has the form:

1) , where f(x) is a function that has derivatives of all orders at x=a. R n - the remainder term in the Taylor series is determined by the expression

2)

The k-th coefficient (at x k) of the series is determined by the formula

3) A special case of the Taylor series is the Maclaurin (=McLaren) series (the expansion occurs around the point a=0)

at a=0

members of the series are determined by the formula

Conditions for using Taylor series.

1. In order for the function f(x) to be expanded into a Taylor series on the interval (-R;R), it is necessary and sufficient that the remainder term in the Taylor (Maclaurin (=McLaren)) formula for this function tends to zero as k →∞ on the specified interval (-R;R).

2. It is necessary that there are derivatives for a given function at the point in the vicinity of which we are going to construct the Taylor series.

Properties of Taylor series.

If f is an analytic function, then its Taylor series at any point a in the domain of definition of f converges to f in some neighborhood of a.

There are infinitely differentiable functions whose Taylor series converges, but at the same time differs from the function in any neighborhood of a. For example:

Taylor series are used in approximation (approximation is a scientific method that consists of replacing some objects with others, in one sense or another close to the original ones, but simpler) of a function by polynomials. In particular, linearization ((from linearis - linear), one of the methods of approximate representation of closed nonlinear systems, in which the study of a nonlinear system is replaced by the analysis of a linear system, in some sense equivalent to the original one.) equations occurs by expanding into a Taylor series and cutting off all terms above first order.

Thus, almost any function can be represented as a polynomial with a given accuracy.

Examples of some common expansions of power functions in Maclaurin series (=McLaren, Taylor in the vicinity of point 0) and Taylor in the vicinity of point 1. The first terms of expansions of the main functions in Taylor and McLaren series.

Examples of some common expansions of power functions in Maclaurin series (=McLaren, Taylor in the vicinity of point 0)

Examples of some common Taylor series expansions in the vicinity of point 1

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 with an indication of 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 such 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) is equal to the 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.

Integration is one of the main operations in mathematical analysis. Tables of known antiderivatives can be useful, but now, after the advent of computer algebra systems, they are losing their significance. Below is a list of the most common primitives.

Table of basic integrals

Another, compact option

Table of integrals of trigonometric functions

From rational functions

From irrational functions

Integrals of transcendental functions

"C" is an arbitrary integration constant, which is determined if the value of the integral at any point is known. Each function has an infinite number of antiderivatives.

Most schoolchildren and students have problems calculating integrals. This page contains integral tables from trigonometric, rational, irrational and transcendental functions that will help in the solution. A table of derivatives will also help you.

Video - how to find integrals

>>Integration methods

Basic integration methods

Definition of integral, definite and indefinite integral, table of integrals, Newton-Leibniz formula, integration by parts, examples of calculating integrals.

Indefinite integral

A function F(x) differentiable in a given interval X is called antiderivative of the function f(x), or the integral of f(x), if for every x ∈X the following equality holds:

F " (x) = f(x). (8.1)

Finding all antiderivatives for a given function is called its integration. Indefinite integral function f(x) on a given interval X is the set of all antiderivative functions for the function f(x); designation -

If F(x) is some antiderivative of the function f(x), then ∫ f(x)dx = F(x) + C, (8.2)

where C is an arbitrary constant.

Table of integrals

Directly from the definition we obtain the main properties of the indefinite integral and a list of tabular integrals:

1) d∫f(x)dx=f(x)

2)∫df(x)=f(x)+C

3) ∫af(x)dx=a∫f(x)dx (a=const)

4) ∫(f(x)+g(x))dx = ∫f(x)dx+∫g(x)dx

List of tabular integrals

1. ∫x m dx = x m+1 /(m + 1) +C; (m ≠ -1)

3.∫a x dx = a x /ln a + C (a>0, a ≠1)

4.∫e x dx = e x + C

5.∫sin x dx = cosx + C

6.∫cos x dx = - sin x + C

7. = arctan x + C

8. = arcsin x + C

10. = - ctg x + C

Variable replacement

To integrate many functions, use the variable replacement method or substitutions, allowing you to reduce integrals to tabular form.

If the function f(z) is continuous on [α,β], the function z =g(x) has a continuous derivative and α ≤ g(x) ≤ β, then

∫ f(g(x)) g " (x) dx = ∫f(z)dz, (8.3)

Moreover, after integration on the right side, the substitution z=g(x) should be made.

To prove it, it is enough to write the original integral in the form:

∫ f(g(x)) g " (x) dx = ∫ f(g(x)) dg(x).

For example:

1)

2) .

Method of integration by parts

Let u = f(x) and v = g(x) be functions that have continuous . Then, according to the work,

d(uv))= udv + vdu or udv = d(uv) - vdu.

For the expression d(uv), the antiderivative will obviously be uv, so the formula holds:

∫ udv = uv - ∫ vdu (8.4.)

This formula expresses the rule integration by parts. It leads the integration of the expression udv=uv"dx to the integration of the expression vdu=vu"dx.

Let, for example, you want to find ∫xcosx dx. Let us put u = x, dv = cosxdx, so du=dx, v=sinx. Then

∫xcosxdx = ∫x d(sin x) = x sin x - ∫sin x dx = x sin x + cosx + C.

The rule of integration by parts has a more limited scope than substitution of variables. But there are whole classes of integrals, for example,

∫x k ln m xdx, ∫x k sinbxdx, ∫ x k cosbxdx, ∫x k e ax and others, which are calculated precisely using integration by parts.

Definite integral

The concept of a definite integral is introduced as follows. Let a function f(x) be defined on an interval. Let us divide the segment [a,b] into n parts by points a= x 0< x 1 <...< x n = b. Из каждого интервала (x i-1 , x i) возьмем произвольную точку ξ i и составим сумму f(ξ i) Δx i где
Δ x i =x i - x i-1. A sum of the form f(ξ i)Δ x i is called integral sum, and its limit at λ = maxΔx i → 0, if it exists and is finite, is called definite integral functions f(x) of a before b and is designated:

F(ξ i)Δx i (8.5).

The function f(x) in this case is called integrable on the interval, numbers a and b are called lower and upper limits of the integral.

The following properties are true for a definite integral:

4), (k = const, k∈R);

5)

6)

7) f(ξ)(b-a) (ξ∈).

The last property is called mean value theorem.

Let f(x) be continuous on . Then on this segment there is an indefinite integral

∫f(x)dx = F(x) + C

and takes place Newton-Leibniz formula, connecting the definite integral with the indefinite integral:

F(b) - F(a). (8.6)

Geometric interpretation: the definite integral is the area of ​​a curvilinear trapezoid bounded from above by the curve y=f(x), straight lines x = a and x = b and a segment of the axis Ox.

Improper integrals

Integrals with infinite limits and integrals of discontinuous (unbounded) functions are called not your own. Improper integrals of the first kind - these are integrals over an infinite interval, defined as follows:

(8.7)

If this limit exists and is finite, then it is called convergent improper integral of f(x) on the interval [a,+ ∞), and the function f(x) is called integrable over an infinite interval[a,+ ∞). Otherwise, the integral is said to be does not exist or diverges.

Improper integrals on the intervals (-∞,b] and (-∞, + ∞) are defined similarly:

Let us define the concept of an integral of an unbounded function. If f(x) is continuous for all values x segment , except for the point c, at which f(x) has an infinite discontinuity, then improper integral of the second kind of f(x) ranging from a to b the amount is called:

if these limits exist and are finite. Designation:

Examples of integral calculations

Example 3.30. Calculate ∫dx/(x+2).

Solution. Let us denote t = x+2, then dx = dt, ∫dx/(x+2) = ∫dt/t = ln|t| + C = ln|x+2| +C.

Example 3.31. Find ∫ tgxdx.

Solution.∫ tgxdx = ∫sinx/cosxdx = - ∫dcosx/cosx. Let t=cosx, then ∫ tgxdx = -∫ dt/t = - ln|t| + C = -ln|cosx|+C.

Solution.

Example3.33. Find .

Solution. =

.

Example3.34 . Find ∫arctgxdx.

Solution. Let's integrate by parts. Let us denote u=arctgx, dv=dx. Then du = dx/(x 2 +1), v=x, whence ∫arctgxdx = xarctgx - ∫ xdx/(x 2 +1) = xarctgx + 1/2 ln(x 2 +1) +C; because
∫xdx/(x 2 +1) = 1/2 ∫d(x 2 +1)/(x 2 +1) = 1/2 ln(x 2 +1) +C.

Example3.35 . Calculate ∫lnxdx.

Solution. Applying the integration by parts formula, we obtain:
u=lnx, dv=dx, du=1/x dx, v=x. Then ∫lnxdx = xlnx - ∫x 1/x dx =
= xlnx - ∫dx + C= xlnx - x + C.

Example3.36 . Calculate ∫e x sinxdx.

Solution. Let us denote u = e x, dv = sinxdx, then du = e x dx, v =∫ sinxdx= - cosx → ∫ e x sinxdx = - e x cosx + ∫ e x cosxdx. We also integrate the integral ∫e x cosxdx by parts: u = e x , dv = cosxdx, du=e x dx, v=sinx. We have:
∫ e x cosxdx = e x sinx - ∫ e x sinxdx. We obtained the relation ∫e x sinxdx = - e x cosx + e x sinx - ∫ e x sinxdx, from which 2∫e x sinx dx = - e x cosx + e x sinx + C.

Example 3.37. Calculate J = ∫cos(lnx)dx/x.

Solution. Since dx/x = dlnx, then J= ∫cos(lnx)d(lnx). Replacing lnx through t, we arrive at the table integral J = ∫ costdt = sint + C = sin(lnx) + C.

Example 3.38 . Calculate J = .

Solution. Considering that = d(lnx), we substitute lnx = t. Then J = .