# Integral Calculus

• I. N. Bronshtein
• K. A. Semendyayev

## Abstract

Primitive function. Given a function y = f(x) of one variable defined in a closed domain, a function F(x) defined in the same domain(1) whose derivative is equal to f(x) (or, that is to say, whose differential is equal to f(x)dx) is called a primitive function of f(x):
$${F}'(x)=f(x)\;\;\;\;\textup{or}\;\;\;\;dF(x)=f(x)dx$$
.

## Keywords

Line Integral Primitive Function Elementary Domain Integral Calculus Definite Integral
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

1. (1).
In certain cases the domain of definition of a primitive function F(x) is wider than that of the given function f(x). If the function f(x) is defined in a connected domain except for isolated points of discontinuity x1,x2,…,xn, then the domain of definition of its primitive function may contain these points (see p. 394).Google Scholar
2. (1).
The area of $$ABCD=\int\limits_{a}^{x}f(x)\;dx$$, see p. 455.Google Scholar
3. (2).
See footnote on p. 393.Google Scholar
4. (1).
The name “primitive function” will be used in the sequel as a synonym of “integral”, but, in the tables, the constant C of integration will be omitted for the sake of brevity.Google Scholar
5. (2).
For tables of elliptic integrals see p. 407.Google Scholar
6. (3).
For the graphical integration (construction of the graph of a primitive function from that of the given function) see pp. 463, 464.Google Scholar
7. (1).
u, v, w are functions of x.Google Scholar
8. (2).
u and v are functions of x.Google Scholar
9. (1).
The numbers A, B,…, L can also be obtained by the method of undetermined coefficients (see p. 151).Google Scholar
10. (1).
The polynomials P1(x) and P2(x) can be found easily, if the factors of P(x) are known, i.e., if all the roots of the equation P(x) = 0 are determined. But P1(x) and P2(x) can also be found without solving this equation: it is sufficient to integrate the polynomial P(x) and find the highest common factor of P(x) and P′(x) (see p. 149). which is equal to P1(x), while $$P_{2}(x)=\frac{P(x)}{P_{1}(x)}$$.Google Scholar
11. (1).
In cases, when the integrals (A) are expressible in terms of elementary functions, they are called pseudo elliptic integrals.Google Scholar
12. (1).
The symbol R denotes a rational function of the expressions to which it refers.Google Scholar
13. (1).
Integrals of functions involving sin x and cos x together with the hyperbolic functions and the function eax are given on pp. 448, 449.Google Scholar
14. (1).
The definite integral $$\int\limits_{0}^{x}\frac{\textup{sin}\;t}{t}\;dt$$ is called the integral sine and denoted by Si x: $$\textup{Si}\;x=x-\frac{x^{3}}{3\cdot 3!}+\frac{x^{5}}{5\cdot 5!}-\frac{x^{7}}{7\cdot 7!}+\cdots$$Google Scholar
15. (2).
Bn are Bernoulli’s numbers (see p. 354).Google Scholar
16. (1).
The definite integral $$\int\limits_{x}^{\infty} \frac{\textup{cos}\;t}{t}\;dt$$ is called the integral cosine and denoted by Ci x: $$\textup{Ci}\;x=C-\textup{ln}\;|x|-\frac{x^{2}}{2\cdot 2!}+\frac{x^{4}}{4\cdot 4!}-\frac{x^{6}}{6\cdot 6!}+\cdots$$, where C is Euler’s constant (see p. 331).Google Scholar
17. (1).
En are Euler’s numbers (see p. 354).Google Scholar
18. (1).
Bn are Bernoulli’s numbers (see p. 354).Google Scholar
19. (1).
Bn are Bernoulli’s numbers (see p. 354).Google Scholar
20. (1).
The definite integral $$\int\limits_{-\infty} ^{x}\frac{e^{t}}{t}\;dt$$ is called the integral exponential function and denoted by Ei x. When x < 0, the integral is divergent for t = 0; in this case Ei x should be understood to be the principal value of the improper integral (see p. 475): $$\int\limits_{-\infty} ^{x}\frac{e^t}{t}\;dt=C+\textup{ln}\;|x|+\frac{x}{1\cdot 1!}+\frac{x}{2\cdot 2!}+\cdots +\frac{x^n}{n\cdot n!}+\cdots$$ (C is Euler’s constant, see p. 331).Google Scholar
21. (1).
The definite integral $$\int\limits_{0}^{x}\frac{dt}{\textup{ln}\;|t|}$$ is called the integral logarithm and denoted by Li x. If x > 1, the integral is divergent for t = 1. In this case, Li x should be understood to be the principal value of the improper integral (see p. 475). The integral logarithm is related to the integral exponential function (see p. 448) by Li x = Ei (ln x ).Google Scholar
22. (1).
Bn are Bernoulli’s numbers (see p. 354).Google Scholar
23. (1).
The concept of the definite integral can also be generalized for functions defined in an arbitrary connected interval (an open interval, an interval open on one side, a half-axis or the whole number axis) or in a connected interval except a finite number of separate points. The integrals considered in this more general sense belong to the improper integrals (see p. 471–478).Google Scholar
24. (1).
The definite integral exists also for any bounded function with a finite number of points of discontinuity in the interval [a, b]. A function for which the definite integral exists in the given interval is called integrable in this interval.Google Scholar
25. (1).
In order to avoid misunderstanding, the variable of integration has been denoted by t (see p. 455).Google Scholar
26. (1).
In this example, we substitute first x = φ(t) = a sin t, whence t = ψ(x) = arc sin (x/a); the function ψ(x) is single-valued in the interval [0, a] and ψ(0) = 0, ψ(a) = ½ π. Then we substitute t = φ(z) = ½ z, whence z = ψ(t) = 2t; this is a single-valued function in the interval [0, ½π] and ψ(0) = 0, ψ (½ π) = π.Google Scholar
27. (1).
In a general case, when the direction of the force does not coincide with the direction of the motion, the work can be calculated as a line integral (see p. 635).Google Scholar
28. (1).
The concept of an integral can also be generalized to more complicated cases, when the domain of definition of the function (the domain of integration) is the set of values of another function (Stieltjes’ integral).Google Scholar
29. (1).
Similarly as in case (1), we assume in integrals (2) and (3) that $$\lim_{x \to a}\;f(x)=\infty$$ or $$\lim_{x \to c}\;f(x)=\infty$$.Google Scholar
30. (1).
For the Gamma function see p. 191; for tables of values of Γ(x), see p. 87.Google Scholar
31. (1).
C is Euler’s constant (see p. 331).Google Scholar
32. (2).
$$B(x,\;y)=\frac{\mathit{\Gamma}(x)\;\mathit{\Gamma}(y)}{\mathit{\Gamma}(x+y)}$$is the so-called Beta function or Euler’s integral of the first kind and Γ(x) is the Gamma function or Euler’s integral of the second kind (see p. 191).Google Scholar
33. (1).
E and K are complete elliptic integrals: E = E(k, ½π), K = F(k, ½π) (see p. 408 and the table on pp. 92, 93).Google Scholar
34. (2).
C is Euler’s constant (see p. 331).Google Scholar
35. (1).
Γ(x) is the Gamma function (see p. 191 and the table on p. 87).Google Scholar
36. (1).
$$B(x,\;y)=\frac{\mathit{\Gamma}(x)\;\mathit{\Gamma}(y)}{\mathit{\Gamma}(x+y)}$$ is the Beta function or Euler’s integral of the first kind and Γ(x) is the Gamma function or Euler’s integral of the second kind (see p. 191).Google Scholar
37. (2).
Γ(x) is the Gamma function (see p. 191 and the table on p. 87).Google Scholar
38. (1).
For connected domains of two variables see p. 341.Google Scholar
39. (1).
Moreover, we assume that to each point of the projection of K onto the x axis, there corresponds a unique point of K (i.e., a point of the curve is uniquely determined by its projection onto the x axis). If this condition is not satisfied, we decompose the arc K into several parts such that each of them satisfies the condition in question; the line integral along the arc K is regarded as the sum of integrals taken along each of these parts of K.Google Scholar
40. (1).
A vectorial exposition of the line integral in the general form and mechanical significance of the line integral is given in the chapter “Field theory”, p. 635.Google Scholar
41. (1).
A primitive function U(x, y) is a potential of the vector field Pi+ Qj (in another notation, it is a potential with the sign “ − ”), see p. 636.Google Scholar
42. (1).
The function u is regarded here as a function of a point (see p. 340) which may be defined not only in Cartesian coordinates.Google Scholar
43. (1).
The requirement that the area dSi tends to zero is not sufficient. It is necessary to require that the diameter of each ΔSi, i.e., the distance between the most remote points of ΔSi tends to zero. For example, if one side of a rectangle tends to zero, its area tends also to zero, but the diameter remains finite.Google Scholar
44. (1).
In the same sense, as for the double integral, i.e., not only the area of ΔVi, but also its diameter tends to zero.Google Scholar
45. (1).
Since, in this case, f(x, y, z) ≡ 1, this integral represents the volume of the solid V.Google Scholar
46. (1).
These integrals are an extension of double integrals (p. 495) just as the line integrals of the first type (p. 486), are an extension of the ordinary definite integrals (p. 454).Google Scholar
47. (2).
In the same sense as in the case of a double integral (see footnote on p. 496).Google Scholar
48. (1).
We assume here that to every point of the projection S′ on the xy plane there corresponds a unique point of S (i.e., that a point of S is uniquely determined by its projection on the xy plane). If the surface S fails to have this property, we decompose it into several parts each of which satisfies this condition and consider the surface integral over S as the sum of the surface integrals taken over all parts of S.Google Scholar
49. (2).
In evaluation of a surface integral of the first type, this angle is always considered as an acute angle; cos γ > 0.Google Scholar
50. (1).
There exist surfaces whose two sides cannot be distinguished (e.g., Möbius band); we do not consider such surfaces in mathematical analysis.Google Scholar
51. (1).
A vectorial treatment of the surface integral in general form is given in the chapter “Field theory” (p. 625).Google Scholar
52. (1).
A vectorial treatment of these theorems is given in the chapter “Field theory” (pp. 645, 646).Google Scholar
53. (2).
This theorem is valid provided that the functions P, Q, R are continuous and have continuous derivatives of the first order.Google Scholar
54. (1).
This theorem is valid provided that the functions P, Q, R are continuous and have continuous derivatives.Google Scholar