Calculus and Probability for Actuarial Students

You are reading a page from Calculus and Probability for Actuarial Students, Alfred Henry (1927)
Part of the American Term Life Insurance History Project
Term Life Insurance

CHAPTER XVII
INTEGRAL CALCULUS. STANDARD FORMS

1. In the preceding chapter we found that the process of integrais the converse of that of differentiation. In other words, that given f' (x) we have to find a function f (x) such that f^' (x) is its first differential coefficient. The analogy with the process of finite integration is apparent and the remarks made in Chapter VII, 5, apply equally to the present case.
It may, therefore, be said again that the process of integration cannot be carried out for every function, that definite rules cannot be laid down to apply in every case, and that the student must look to applying the results of the differential calculus as a guide in solving any problem presented by the function under consideration.
2. In the first place it is necessary to point out that the ordinary algebraic laws apply to the integrating symbol fdx in the same way as they have been shown to apply to the symbolic operations 0 and d
_d- _x. Thus:

The operation is distributive for, if u, v, w, ... be any funcof x,

flf

udx+fvdx+Jwdx+...}=u+v+w...

and therefore

rudx+ fvdx+ fwdx+...=J(u+v+w+..•)dx ...(I).

(2)The operation is commutative with regard to constants, for,

if da=u,    d(cv)_cdv__cu,
dx    dv    dx
therefore    fcudx=cv_^—cludx     (2).
3. It should be added that the process of integration introduces a constant into every indefinite integral, for, if u = d_x,
d (v + c) _ dv _
dx    dx    '

I

94    INTEGRAL CALCULUS
and therefore    fudx = v + c,
which result is, indeed, obvious from the consideration that a constant term disappears on differentiation.

4.In accordance with the symbolic notation which has been previously developed (Chapter X`', 1), we may write

^{(x) dx}

=(d)-lf(x)

(3)

-'.f

(

)

[loge( + O)

(4).

5.Standard Forms.

ⁿ

(a) x'

Since

d \

'_±

1 =

_xn

n+1

xn+i

therefore

ix"

dx = n

(b)x^—'. Sinced (log x) _ 1

dxx'

therefore

j- f

log

+ c.

(c)e^x. Since^d (ex) = c^x

dx'

thereforefedx = e^z + c. d ax l

(d)

Since\

log,

log

_axe

log

therefore

dx =

_lo

^ge

Other integrals may be obtained from a consideration of the table

of standard forms of differential coefficients given in Chapter XI,

STANDARD FORMS 95
3. The results are embodied in the following table. Their verifiis left as an exercise to the student.

Function	Integral
rx"dx ! ¹dx x Ja^xdx f e^xdx f cosxdx yin x dx f sec²xdx Icosec²xdx	xⁿ⁺ⁱn 1 log,x a^xlug, a e^xsinx — cos x tanx —cotx

In each case the constant c has been omitted for the sake of simplicity, but its importance in the result must not be forgotten.

Sums of the above functions can be integrated by the use of the distributive property of the operation of integration. Thus

(x + 2

) dx

x dx +12

=~x2+loge

₂

+C.

Similarly

J (ax"

+ bx-

⁴

)

⁴

dx + b

x ' dx

^—3

+C.

The problem of integration may be presented in another way. Thus to find the sum of

⁺

x+na

⁺

x+2m

⁺⁺

x+xm

when x is indefinitely increased.

96    INTEGRAL CALCULUS
The series may be written as
1 ^rl    1    1    1    ' m
m 1    in⁺ tin⁺⁺    m x
1+— 1+    l+x.
x    x    x

The terms in the denominator consist of values of a quantity, which

can be represented by y, increasing by equal increments of ^m. Also x
the sum of the series is multiplied by —^m, which is the increment x
in the value of the denominator.
Since the initial value of the denominator y is unity and the final value is 1 + m, from our definition of an integral we may write,
when x-.-co and therefore ^m-..O,
x
Lt ^{1 rl}+ ¹ + 1 + ... +    1    m _ 1 I+m dy
_    _ _ f _
    nt 1 1+^m l+ ~ⁿ    l+x.in x in    ^yx    x    x,
1
= m [log i+m
= ma log 1 +m.