4.6 Integration of certain other Classes of Functions
4.6.1 Preliminary Remarks on the Rational Representation of the Trigonometric and Hyperbolic Functions: The integration of certain other general classes of functions can be reduced to the integratioa of rational functions. We shall better understand this reduction, if we begin by stating certain elementary facts about the trigonometric and hyperbolic functions. If we set t = tanx/2, elementary trigonometry yields the simple formulae
in fact,
whence we obtain the above equations from the elementary formulae
These equations show that sin x and cos x can be expressed rationally in terms of the quamtity t=tanx/2. Differentiation now yields
whence
whence the derivative dx/dt is also a rational expression in t.
Fig. 2 shows the geometrical representation and
the geometrical meaning of our 
formulae.
It shows the circle u² + v² = 1 in a
uv-plane. If x denotes the angle POT,
then u = cos x and v = sin x.
By a theorem in elementaly geometry, the angle OSP with
its vertex at the point u=-1, v=0 is equal to x/2,
and we can deduce from the figure the geometrical meaning of the
parameter:
t = tan x/2 = OR.
If the point P starts from S and moves once around the circle in the positive direction, i.e., if x moves through the interval from -p to +p, the quantity t will move exactly once throngh the entire range of the values from - ¥ to +¥.
We may correspondingly express the hyperbolic functions cosh x = (ex+e-x)/2 and sinh x=(ex-e-x)/2 as rational functions of a third quantity. The most obvious way is to set ex = t, so that we have for these hyperbolic functions the rational expressions
Once again, dx/dt = 1/t. However, we obtain a closer analogy with the trigonometric functions by introduction of the quantity t = tanh x/2; we then arrive at the formulae
Differentiation of t = tanh x/2 yields this time the rational expression
for the derivative dx/dt. Once again, the quantity t has a geometrical meaning similar to that which it has in the case of the trigonometric functions, as we see at once from Fig. 3.
However, while in the case of the trigonometrio functions, t must run through the entire range of values from - ¥ to +¥, in order to yield all pairs of values of cos x and sin x, in the case of the hyperbolic functions, t is limited to the interval -1 < t < l.
After these preliminary remarks, we turn to our integration problem.
4.6.2 Integration of R(cos x, sin x): Let R(cos x, sin x) denote an expression which ia rational in the two functions sin x and cos x, i.e., an expression which is formed rationally from these two functions and given constants such as
If we apply the substitution t = tan x/2, the integral
becomes the integral
and we have now under the integral sign a rational function of t. Thus, we have obtained theoretically the integral of our expression, since we can now perform the integration by the methods of the preceding section.
4.6.3 Integration of R(cosh x, sinh x): In the same way, if R(cosh x, sinh x) is an expression which is rational in terms of the hyperbolic functions cosh x and sinh x, we can integrate it by means of the substitution t = tanh x/2. Recalling that
we find
(According to a previous remark, we could also have introduced t = ex as a new variable and expressed cosh x and sinh x in terms of t.) The integration is once again reduced to that of a rational function.
4.6.4 lntegration of 
: The integral can be reduced to
the type in 4.6.2 by using the substitution
the transformation t = tan x/2 leads to the integration of a rational function. By the way, we could have carried out the reduction in a single step by the substitution
that is, we could have obtained directly a rational function by introduction of a new variable t=tanu/2.
4.6.5 Integration of 
: The integral of this function is transformed by the
substitution x = cosh u into the type 4.6.3,
when we again introduce
4.6.6
Integration of 
: The integral of
this function is transformed by the substitution x =
sinh u into the type
4.6.3, whence it can be integrated in
terms of elementary functions. Instead of the further reduction
to an integral of a rational function by the substitution eu=t or tanh u/2
= t, we could have reached the integral of a rational
function by a single step by either of the substitutions
4.6.7 Integration of 
:The integral of this function is rational in terms of x
and the square root of an arbitrry polynomial of the second
degree in x; it can be reduced immediately to one of the
types above. As in 4.5.1,we write
If ac - b² > 0, we
introduce a new variable x = by means of the transformation 
whence the surd takes
the form 
. Expressed in terms of x , we have now the
type of 4.6.6. The constant a must be positive here, in
order that the square root may have real values.
If ac - b² = 0, a > 0, then we see from
that the integrand was rational from the start.
Finally, if ac - b² < 0,
we set 
and obtain for the surd the expression 
. If a
is positive, our integral is thus reduced to the type 4.6.5; on the other hand, if a is negative, we write
the surd in the form 
and see that the integral is thus
reduced to the type of 4.6.4
4.6.8 Further Examples of Reduction to Integrals of Rational Functions: We shall briefly mention two other types of functions which can be integrated by reduction to rational functions:
(1) Rational expressions involving two different surds of linear expressions
(2) Expressions of the form
where a, b, a, b are
constants. In (1), we introduce the new variable 
so that
ax+b=x², whence
then
which is of the type 4.6.7.
If we introduce in the second case the new variable
we have
and immediately arrive at
which is an integral of a rational function.
4.6.9 Comments on Exercises: The preceding discussion is mainly of theoretical interest. In the case of complicated expressions, the actual calculations would be far too involved. It is therefore expedient to employ, whenever it is possible, a special form of the integrand, in order to simplify the work. For example, in order to integrate the expression
1/(a²sin² x + b²cos²x),
it is better to use the substitution t = tan x instead of that given in 4.6.2, since sin² x and cos² x can be expressed rationally in terms of tan x and it is therefore unnecessary to go back to t = tan x/2. The same is true for every expression formed rationally from sin²x, cos² x and sin x cos x*, because sin x cos x = tan x cos² x can, of course, be expanded in terms of tan x. Moreover, for the calculation of many integrals, a trigonometrical form is to be preferred to a rational one, provided that the trigonometrical form can be evaluated by some simple recurrence method.
For example, although the integrand in 
can be
reduced to a rational form, it is simpler to write x =
sin u and convert it into 
since this can be
treated easily by the recurrence method in 4.4.3 (or by using the addition theorems to reduce the powers
of the sine and cosine to sines and cosines of multiple angles).
For the evaluation of the integral
we determine, instead of referring to the general theory, a number A and an angle q in such a way that
that is, we write
The integral then becomes
and, on introducing the new variable x+q, (4.2.3) yields the value of the integral
Evaluate the integrals
4.7 Remarks on Functions which are not Integrable in Terms of Elementary Functions
4.7.1 Definition of Functions by means of Integrals. Elliptic Integrals: The above examples of types of functions which can be integrated by reduction to rational functions practically exhaust the list of such functions. Attempts to express general integrals such as
or 
in terms of elementary functions have
always ended up in failure; and finally, in the Nineteenth
Century, it has been proved that it is actually impossible to
cany out integrations in terms of elementary functions. Hence, if
the objective of the integral calculus were integration of
functions in terms of elementary functions, we should have come
to a definite halt. But such a restricted objective has no
intrinsic justiflcation; indeed, it is of a somewhat artificial
nature. We know that the integral of every continuous function
exists and is itself a continuona function of the upper limit,
and this fact has nothing to do with the question whether an
integral can be expressed in terms of elementary functions or
not. The distinguishing features of the elementary functions are
based on the fact that their properties are easily recognized,
that their application to numerical problems is often facilitated
by convenient tables or, as in the case of the rational
functions, that they are readily calculated with as great a
degree of accuracy as we please.
Whenever an integral of a function cannot be expressed in terms of functions with which we are already acquainted, there is nothing to stop us from introducing such an integral as a new higher function into analysis, which really means no more than giving it a name. Whether the introduction of such a new function is convenient or not depends on the its properties, the frequency at which it occurs and the ease with which it can be manipulated in theory and in practice. Hence, in this sense, the process of integration forms a base for the generation of new functions.
After all, we are already acquainted with this principle from our dealings with the elementary functions. Thus, we found ourselves obliged in 3.6 to introduce the previously unknown integral of 1/x as a new function, which we called the logarithm and the properties of which we could easily determine. We could have introduced the trigonometric functions in a similar way, only making use of the rational functions, the process of integration, and the process of inversion. For this purpose, we only need introduce one or the other of the equations
as the definitions of the function artan x or arsin x, respectively, in order to arrive at the trigonometric functions by inversion. By this process, the definition of these functions becomes separated from geometry, but we are naturally left with the task of also developing their properties independently of geometry.
We shall not go here into the development of these ideas. The essential step is to prove the addition theorem for the inverse functions, i,e. for sin and tan.
The first and most important example, which leads us beyond the region of elementary functions, is are the elliptic integrals. These are integrals in which the integrand is formed in a rational way from the variable of integration and the square root of an expression of the third or fourth degree. Among these integrals, the function
turns out to be of special importance. Its inverse function s(u) has a correspondingly important role. In particular, for k = 0, we obtain u(s) = arsin x and s(u) = sin u, respectively. The function s(u) has been examined as thoroughly and tabulated as the elementary functions. However, this leads us away from the line of the present discussioa and into the realm of the so-called elliptic functions, which occupy a central position in the theory of functions of a complex variable. We shall merely note here that the name elliptic integral arises from the fact that such integrals enter into the problem of the determination of the length of an arc of an ellipse.(Chapter V)
Moreover, we may note that integrals which at first sight have quite a different appearance turn out after a simple substitution to be elliptic integrals. For example, the integral
is transformed by the substitution u = cos x/2 into the integral
the integral
by the substitution u = sin x into the integral
and, finally, the integral
by the substitution u = sin x into the integral
4.7.2 On Differentiation and Integration: We insert here another remark on the relationship between differentiation and integration. Differentiation may be viewed to be a more elementary process than integration, since it does not lead us away from the domain of known functions. On the other hand, we must remember that the differentiability of an arbitrary, continuous function is by no means a foregone conclusion, but a very stringent additional assumption. In fact, we have seen that there are continuous functions which are not differentiable at isolated points and we may mention without proof that, since the time of Weierstrass, many examples have been constructed of continuous functions which do not possess a derivative anywhere (cf. Titchmarsh, The Theory of Functions (Oxford, 1932), §§ 11.21 - 11.23, pp/ 350- 354.) Hence, there is much less in the mathematical definition of continuity than simple intuition would lead us to suspect. In contrast, even though integration in terms of elementary functions is not always possible, we are under all circumstances at least certain that the integral of a continuous function exists.
Altogether, we see that integration and differentiation cannot be simply classified as more elementary and less elementary, but that from some points of view the one and from other points of view the other should be considered to be more elementary.
In as far as the concept of integral is concerned, we shall see in the next section that it is not closely tied to the assumption that the integrand is continuous, but that it may be extended to wider classes of functions with discontinuities.
4.8 Extension of the Concept of Integral. Improper Integrals
4.8.1 Functions with Jump
Discontinuities. To start
with, we see that there arises no 
difficulty with an extension of the concept of integral
to the case where the integrand has jump discontinuities at one
or more points in the interval of integration. In fact, we need
only take the integral of the function as the sum of the
integrals over the separate subintervals in which the function is
continuous. The integral then retains its intuitive meaning as an
area (Fig. 4).
We should really note that, in our previous definition of integral, we assumed that the interval is closed and tbe function is continuous in the closed interval. This presents no difficulties, since we can extend in each closed subinterval the function so that it is continuous by taking for the value of the function at the end-point the limit of the function as x approaches the end-point from inside the interval.
4.8.2 Functions with Infinite Discontinuities: It is quite a different matter when a function has an infinite discontinuity inside the interval or at one of its ends. In order to gain even a notion of an integral in this case, we must introduce a further limiting process. Before stating the general definition, we shall illustrate some of the possibilities by means of examples.
We begin with the integral
where a is a positive number. The integrand 1/xa becomes infinite as x ® 0, whence we cannot extend the integral to the lower limit 0. However, we can try to find what happens as we take the integral from the positive limit e to the limit 1, say, and finally let e ® 0. According to the elementary rules of integration, we obtain, provided a ¹ 1,
We recognize immediately that the following possibilities arise: (1) a > 1, when, as e ® 0, the right hand side tends to ¥; (2) a < 1, when the right hand side tends to the limit 1/(1 - a). hence, in the second case, we shall simply take this limiting value as the integral between the limits 0 and 1. In the first case, we shall say that the integral from 0 to 1 does not exist. (3) In the third case, where a = 1, the integral will be equal to -log e and therefore, as e ® 0, it approaches no limit, but tends to ¥, that is, the integral from 0 to 1 does not exist.
Another example of the extension of the
integral of a function up to an infinite discontinuity is given
by the integrand 
We find that
If we let e tend to 0, the right hand side converges to the
definite limit p/2; we therefore call this the value of the integral 
, even
though the integrand becomes infinite at the point x =
1.
In order to extract a perfectly general concept from these examples, we note, in the first place, that it clearly makes no essential difference whether the discontinuity of the integrand lies at the upper or the lower end of the interval of integration. We now make the following statement:
If in an interval a £ x £ b
the function f(x) is continuous with the single
exception of the end-point b, we define 
as the
limit
when the point b - e approaches the end-point b from inside the interval provided such a limit exists.
In this case, we say that the improper integral 
converges. However,
if no such limit exists, we say that the integral does not exist or does not converge or
that it diverges.
An analogous definition holds for the case where the lower limit of the interval of integration instead of the upper one is the exceptional point.
Even improper integrals can be intepreted as areas. In the first instance, of course, there is no sense in speaking of the area of a region which estends to infinity; yet, one may attempt to define such an area by means of a passage to the limit from a bounded region with a finite area. For example, the above results for the function 1/xa imply that the area, bounded by x axis, the line x=e and the curve y = 1/xa tends to a finite limit as e ® 0, provided that a < 1, and that it tends to infinity if a ³ 1. This fact may be simply expressed as follows: The area between the x-axis, the y-axis, the curve and the line x = 1 is finite or infinite according to whether a < 1 or a ³ 1.
Naturally, intuition can give us no precise information about the finitenees or infiniteness of the area of a region stretching to infinity. We can only say about such a region that the more closely its sides approach one another, the more likely it is to have a finite area. In this sense, Fig. 6 illustrates the fact that for a < 1, the area under our curve remains finite, while it is infinite for a ³ 1.
In order to discover whether a function f(x), which has an infinite discontinuity at the point x = b, can be integrated up to b, we can often save ourselves a special investigation by means of the criterion:
Let the function f(x) be
positive (A8.3); we will show that this
restriction of sign can be removed) in the interval a £ x £ b and
let 
Then the integral
converges, if there exist both a positive
number m less than 1 and a fixed number M independent
of x such that everywhere in the interval a £ x £ b the
inequality f(x) £ M/(b - x)m
is true, in other words, if
at the point x = b, the function f(x)
becomes infinite at a lower order at least than the first. On the other hand, the integral diverges, if there
exist both a number n ³
1 and a fixed number N such that
everywhere in the interval a £ x £ b the
inequality f(x) ³ N/(b - x)n
is true, in other words, if
at the point x = b the function f(x)
becomes infinite to the first order at least.
The proof follows almost immediately from a comparison with the very simple special case discussed above. In order to prove the first part of the theorem, we observe that for 0 < e < b - a, we have
As e ® 0, the integral on the right hand side, which is
obtained from the integral 
(cf. 2.7.2), by a simple change of
notation, has a limit and therefore remains bounded. Moreover,
the values of 
increase monotonically as e ® 0; since they
are also bounded, they must possess a limit and therefore the
integral converges.
The parallel proof of the second part of the theorem is left as an exercise for the reader.
We likewise see at once that exactly analogous theorems hold when the lower limit of the integral is a point of infinite discontinuity. If a point of infinite discontinuity lies in the interior of the interval of integration, we merely use this point to subdivide the interval into two parts and then apply the above considerations separately to each of these.
As a further example, we consider the elliptic integral
We conclud at oncee from the identity 1 - x² = (1 - x)(l + x) that, as x ® 1, the integrand becomes only infinite at order ½, whence the improper integral exists.
4.8.3 Infinite Interval of Integration: Another important extension of the concept of integral occurs when one of the limits of integration as infinite. In order to make this extension precise, we introduce the notation: If the integral
where a is fixed, tends to a definite limit when A increases positively beyond all bounds, we denote the limit by
and call it the integral of the function f(x) from a to ¥ . Of course, such an integral does not necessarily exist or, as we often will say, converge.
Simple examples of the various possibilities are again given by the function f(x)=1/xa :
We see here that, if we again exclude the case a = 1, the integral to infinity exists for the case a > 1, and, in fact,
in contrast, when a < 1, the integral no longer exists. In the case a = 1, the integral again clearly fails to exist, since log x tends to infinity with x. Hence we see that, as regards integration over an infinite interval, the functions 1/xa do not behave in the same way as for integration up to the origin. This statement also is made plausible by a glance at Fig. 5. In , we see that, the larger is a, the more slowly do the curves draw in towards the x-axis when x is large, so that we can readily assume that the area under consideration tends to a definite limit for sufficiently large values of a.
The following criterion for the existence of an integral with an infinite limit is often useful. We again assume that for sufficiently large valued of x, say, for x³a, the integrand has always the same sign, which, without loss of generality, we can choose to be positive. (As we shall see in Chapter 8, this restriction of the sign is easily removed.) Then we have the statement:
The integral 
converges if the function f(x) vanishes at
infinity to a higher order than the first, that is, if there is a
number n > 1 such that for all values of x,
no matter how large, the relation 0<f(x)£M/xn is true, where M is a fixed
number independent of x. Again, the integral diverges,
if the function remains positive and vanishes at infinity to an
order not higher than the first, that is, if there is a fixed
number N > 0 such that xf(x) ³ N.
The proof of these criteria, which run exactly parallel to the previous argument, is left to the reader.
A very simple example is the integral 
The
integrand vanishes at infinity to the second order. As a matter
of fact, we see at once that the integral does converge, because 
whence
Another equally simple example is
4.8.4 The Gamma Function: Another example of particular importance in analysis is offered by the socalled G-function
Here too the criterion of convergence is
satisfied; for example, if we choose n=2, we have 
since
the exponential function ex tends to
zero to a higher order than any power l/xm
(m > 0). This gamma function, which we can think of
as a function of the number n (not necessarily an
integer), satisfies a remarkable relation, which we can arrive at
in the following way by integration by parts: To begin
with, we have
If we take this formula between the limits 0 and A and then let A increase beyond all bounds, we immediately obtain
and by this recurrence formula, provided m is an integer and 0 < m < n,
In particular, if n is a positive integer, we have
and, since
it follows finally that
This expression for a factorial by an integral is of importanoe in many applications.
The integrals
also converge, as we may easily verify by means of our criterion.
4.8.5 The Dirichlet Integral: A convergent integral, important in many applications, the convergence of which does not follow directly from our criterion and which is a simple case of a type investigated by Dirichlet, is
This integral is easily seen to be convergent, if the upper limit is finite, because sin x/x ® 1 as x ® 0. Its convergence in the infinite interval is due to the periodic change of sign of the integrand, which causes the contributions to the integral from neighbouring intervals of length p almost to cancel each another. In order to utilize this fact, we write the expression
in the form
introduce in the last of the three integrals the new variable x = t - p, whence sin t = -sin x, and obtain
Addition of this expression to the original one for DAB yields
Hence, if we assume that B > A > 0, it follows that
in fact, we may use the method of 2.2.6, while observing that
and
for positive values of x. The integral on the right hand side converges, by our criterion and our formula shows that |DAB| ® 0 as both A and B tend to infinity. Now
and it follows from Cauchy's convergence test that D0B tends to a definite limit as B ® ¥. In other words, the integral I exists. Another proof of this result is given in the Chapter VIII and, moreover, in Chapter IX we shall show that I has the value p/2.
4.8.6
Substitution: It is obvioua
that all rules for the substitution of new variables, etc.
remains valid for convergent
improper integrals. As an example,
in order to calculate 
, we introduce the new variable u
= x² and obtain
Other examples of the use of substitution in the investigation of improper integrals are the Fresnel integrals, which occur in the theory of diffraction of light:
The substitution x² = u yields
Integrating by parts, we have
As A and B tend to ¥, the first two terms on the right hand side tend to 0 and, by the criterion of 4.8.4, the integral also tends to 0. Hence, by the same argument as for the Dirichlet integral, we see that the integral F1 converges. The convergence of the integral F2 is proved in exactly the same way.
These Fresnel integrals show that an improper integral may exist even although the integrand does not tend to zero as x ® ¥. In fact, an improper integral can exist even when the integrand is unbounded, as is shown by the example
When u4 = np, i.e.
when 
n = 0, 1, 2, ··· , the integrand becomes 
, so
that the integrand is unbounded. However, the substitution u²
= x reduces the integral to
which we have just shown to be convergent.
By means of a substitution, an improper integral may often be transformed into a proper one. For example, the transformation x = sin u yields
On the other hand, integrals of continuous functions may be transformed into improper integrals; this oocurs if the transformation u = f (x) is such that at the end of the interval of integration the derivative f '(x) vanishes, so that dx/du is infinite.
Test the convergence of the improper integals 1-11:
12.* Prove that 
does not exist.
13.* Prove that 
14. For what value of s converges (a)
(b)
.
15.* Does 
converge?
16.* (a) If a is a fixed positive number, prove that
(b) If f(x) is continuous in the interval -1 £ x £ 1, prove that
Evaluate the integrals in 1 - 10:
11.* Prove that the substitution x = (a t + b)/(g t+ d), where ad - gb ¹ 0, transforms the integral
into an integral of a similar type, and that, if the biquadratic
has no repeated factors, neither has the new biquadratic in t which takes its place.
Prove that the same statements are true for
where R is a rational function.
The Second Mean Value Theorem of the Integral Calculus
The method of integration by parts yields us an easy way for proving an important theorem on the estimation of integrals, usually called the second mean value theorem of the integral calculus.
Let the function f (x) be monotonic and continuous in the interval a £ x £ b, the derivative f '(x) be continuous and moreover f(x) be an arbitrary function which is continuous in the same interval. Then one has the second mean value theorem of the integral calculus:
There exists a number x such that a £ x £ b for which
We prove this by noting first that we can assume that f (b) = 0; in fact, replacement of f (x) by f (x) - f (b) changes both sides of the equation by the same amount and yields a function which vanishes at x = b. Moreover, we can assume that f (a) > 0; in fact, if f (a) < 0, we need only replace f (x) by -f(x), which changes the sign of both sides of the equation. (The case f (a) = 0 is trivial; in fact, if both f (a) and f (b) vanish, f (x) must be identically zero and our equation becomes 0 = 0.) Hence, we must only prove that, if f (x) is continuous and monotonic decreasing and f (b) = 0, then
We now set 
and apply the formula
for integration by parts to the left hand side of the last
equation; we then have
The integrated part vanishes, since F(a) and f (b) are zero. The expression -f '(a) is everywhere positive, so that we can apply the first mean value theorem of the integral calculus. We thus find that the integral on the right hand side has the value
However,
and our theorem is established.
This theorem can be extended (although we shall not carry out the proof) to more general classes of functions. The theorem remains true for all continuous monotonic functions f (x) whether they have derivatives or not. In fact, it is true for any discontinuous monotonic function for which we are in a position to integrate f(x)f (x).