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The present section is devoted to an introductory discussion of that problem.5.7.1 The Primitive and the Indefinite IntegralDefinition 1 A function F (x) is a primitive of a function f (x) on an interval if Fis differentiable on the interval and satisfies the equation F (x) = f (x), or, what isthe same, dF (x) = f (x) dx.Example 1 The function F (x) = arctan x is a primitive of f (x) =real line, sincearctan x=1.1+x 211+x 2on the entire1Example 2 The function F (x) = arccot x1 is a primitive of f (x) = 1+x2 on the setof positive real numbers and on the set of negative real numbers, since for x = 0111=F (x) = −·−= f (x).1 22x1 + x21 + (x )What is the situation in regard to the existence of a primitive, and what is the setof primitives of a given function?In the integral calculus we shall prove the fundamental fact that every functionthat is continuous on an interval has a primitive on that interval.We present this fact for the reader’s information, but in the present section weshall essentially use only the following characteristic of the set of primitives of agiven function on an interval, already known to us (see Sect.

5.3.1) from Lagrange’stheorem.Proposition 1 If F1 (x) and F2 (x) are two primitives of f (x) on the same interval,then the difference F1 (x) − F2 (x)is constant on that interval.The hypothesis that F1 and F2 are being compared on a connected interval isessential, as was pointed out in the proof of this proposition. One can also see thisby comparing Examples 1 and 2, in which the derivatives of F1 (x) = arctan x andF2 (x) = arccot x1 agree on the entire domain R\0 that they have in common. However,F1 (x) − F2 (x) = arctan x − arccot1= arctan x − arctan x = 0,x5.7 Primitives307for x > 0 while F1 (x) − F2 (x) ≡ −π for x < 0.

For if x < 0, we have arccot x1 =π + arctan x.Like the operation of taking the differential, which has the name “differentiation” and the mathematical notation dF (x) = F (x) dx, the operation of finding aprimitive has the name “indefinite integration” and the mathematical notationf (x) dx,(5.166)called the indefinite integral of f (x) on the given interval.Thus we shall interpret the expression (5.166) as a notation for any of the primitives of f on the interval in question.In the notation (5.166) the sign is called the indefinite integral sign, f is calledthe integrand, and f (x) dx is called a differential form.It follows from Proposition 1 that if F (x) is any particular primitive of f (x) onthe interval, then on that intervalf (x) dx = F (x) + C,(5.167)that is, any other primitive can be obtained from the particular primitive F (x) byadding a constant.If F (x) = f (x), that is, F is a primitive of f on some interval, then by (5.167)we have(5.168)d f (x) dx = dF (x) = F (x) dx = f (x) dx.Moreover, in accordance with the concept of an indefinite integral as any primitive, it also follows from (5.167) thatdF (x) =F (x) dx = F (x) + C.(5.169)Formulas (5.168) and (5.169) establish a reciprocity between the operations ofdifferentiation and indefinite integration.

These operations are mutually inverse upto the undetermined constant C that appears in (5.169).Up to this point we have discussed only the mathematical nature of the constant Cin (5.167). We now give its physical meaning using a simple example. Suppose apoint is moving along a line in such a way that its velocity v(t) is known as afunction of time (for example, v(t) ≡ v). If x(t) is the coordinate of the point attime t, the function x(t) satisfies the equation ẋ(t) = v(t), that is, x(t) is a primitiveof v(t). Can the position of a point on a line be recovered knowing its velocityover a certain time interval? Clearly not.

From the velocity and the time intervalone can determine the length s of the path traversed during this time, but not theposition on the line. However, the position will also be completely determined if itis given at even one instant, for example, t = 0, that is, we give the initial condition3085Differential Calculusx(0) = x0 . Until the initial condition is given, the law of motion could be any lawof the form x(t) = x̃(t) + c, where x̃(t) is any particular primitive of v(t) and c isan arbitrary constant.

But once the initial condition x(0) = x̃(0) + c = x0 is given,all the indeterminacy disappears; for we must have x(0) = x̃(0) + c = x0 , that is,c = x0 − x̃(0) and x(t) = x0 + [x̃(t) − x̃(0)]. This last formula is entirely physical,since the arbitrary primitive x̃ appears in it only as the difference that determinesthe path traversed or the magnitude of the displacement from the known initial pointx(0) = x0 .5.7.2 The Basic General Methods of Finding a PrimitiveIn accordance with the definition of the expression (5.166) for the indefinite integral, this expression denotes a function whose derivative is the integrand. From thisdefinition, taking account of (5.167) and the laws of differentiation, one can assertthat the following relations hold:a.αu(x) + βv(x) dx = α u(x) dx + β v(x) dx + c.

(5.170)b.(uv) dx =u (x)v(x) dx +u(x)v (x) dx + c.(5.171)c. Iff (x) dx = F (x) + con an interval Ix and ϕ : It → Ix is a smooth (continuously differentiable) mappingof the interval It into Ix , then(5.172)(f ◦ ϕ)(t)ϕ (t) dt = (F ◦ ϕ)(t) + c.The equalities (5.170), (5.171), and (5.172) can be verified by differentiating theleft- and right-hand sides using the linearity of differentiation in (5.170), the rulefor differentiating a product in (5.171), and the rule for differentiating a compositefunction in (5.172).Just like the rules for differentiation, which make it possible to differentiate linear combinations, products, and compositions of known functions, relations (5.170),(5.171), and (5.172), as we shall see, make it possible in many cases to reduce thesearch for a primitive of a function either to the construction of primitives for simpler functions or to primitives that are already known.

A set of such known primitives can be provided, for example, by the following short table of indefinite integrals, obtained by rewriting the table of derivatives of the basic elementary functions5.7 Primitives309(see Sect. 5.2.3):x α dx =1x α+1 + cα+1(α = −1),1dx = ln |x| + c,xa x dx =1 xa +cln a(0 < a = 1),ex dx = ex + c,sin x dx = − cos x + c,cos x dx = sin x + c,1dx = tan x + c,cos2 x1sin2 x1dx = − cot x + c,arcsin x + c,dx =√− arccos x + c̃,1 − x21arctan x + c,dx =2− arccot x + c̃,1+xsinh x dx = cosh x + c,cosh x dx = sinh x + c,1cosh2 x1dx = tanh x + c,dx = − coth x + c,sinh2 x#1dx = lnx + x 2 ± 1 + c,√x2 ± 111 1 + x + c.lndx=2 1 − x 1 − x2Each of these formulas is used on the intervals of the real line R on which the corresponding integrand is defined. If more than one such interval exists, the constantc on the right-hand side may change from one interval to another.3105Differential CalculusLet us now consider some examples that show relations (5.170), (5.171) and(5.172) in action.

We begin with a preliminary remark.Given that, once a primitive has been found for a given function on an intervalthe other primitives can be found by adding constants, we shall agree to save writingbelow by adding the arbitrary constant only to the final result, which is a particularprimitive of the given function.a.

Linearity of the Indefinite IntegralThis heading means that by relation (5.170) the primitive of a linear combinationof functions can be found as the same linear combination of the primitives of thefunctions.Example 3a0 + a1 x + · · · + an x n dx == a01 dx + a1x dx + · · · + anx n dx =11an x n+1 .= c + a0 x + a1 x 2 + · · · +2n+1Example 4 √1 21x+√x2 + 2 x +dx =dx =xx121/2= x dx + 2 x dx +dx =x14= x 3 + x 3/2 + ln |x| + c.33Example 511(1 + cos x) dx =(1 + cos x) dx =2211111 dx +cos x dx = x + sin x + c.=2222xdx =cos22b.

Integration by PartsFormula (5.171) can be rewritten asu(x)v(x) = u(x) dv(x) + v(x) du(x) + c5.7 Primitives311or, what is the same, asu(x) dv(x) = u(x)v(x) − v(x) du(x) + c.(5.171 )This means that in seeking a primitive for the function u(x)v (x) one can reducethe problem to finding a primitive for v(x)u (x), throwing the differentiation ontothe other factor and partially integrating the function, as shown in (5.171 ), separating the term u(x)v(x) when doing so. Formula (5.171 ) is called the formula forintegration by parts.Example 6ln x dx = x ln x −x d ln x = x ln x −x·= x ln x −Example 71 dx = x ln x − x + c.x e dx =2 x1dx =xx de = x e −22 xx= x 2 ex − 2e dx = x e − 2x22 xxex dx =x dex = x 2 ex − 2 xex − ex dx == x 2 ex − 2xex + 2ex + c = x 2 − 2x + 2 ex + c.c.

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