An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

184

The definite and the indefinite integral.

Bk. III. ch. I.

c) Let us introduce a new variable for x into the formula
F(x) f(x) dx
by the equation x -- p((), such that to continuous consecutive values
of x from a to b correspond uniquely continuous consecutive values
of ut not undergoing any alternation of increase or decrease, so that
du does not vanish: thus let F(x) become V (u), and let f(x) become, (ut). We have then the relation:
77'(x) f x) V (t dV (u) du
F ~ ( u *~ Z dx'
and so:
dW(t), dx
d d( (u) u (u),
whence:
v () = (u) ' (u ) d ) u.
Substitution of a variable by the equation x =p(u)
reduces the determination of the integral of f:
f(x)dx, to that of: f'() ()'(u)p) du;
by an apt choice of the formula of substitution this integral
may be more simply found than the original one.
When the original is to be taken between the limits x == a and
x = b, the new integral is to be formed with the limits u, and ubc for
which a == (pT(), b = = (uT(), so that the values from x = a to x = b
are uniquely related to the values from Ua to Uzb.
But if the relation between x and u is not uniquely convertible,
the total interval must be broken up into partial intervals in which
a mutually unique relation can be established.
00
Thus, when the integral ex. gr. /f (ax +- b) cx is proposed,
o
a and b being positive, if we put ax + -- = iz, to each value of x
corresponds a unique value of i, but while x passes from 0 to oo, t
which began by decreasing, undergoes a change and subsequently
increases; to each value of in correspond two values of x,
X 1+,~/n2_ 4ab l ddux (1 + ~ )
2a - a 2a~ - 4- -b
as we can realise geometrically by drawing the hyperbola. We can cald u b
culate the minimum value of i by means of d =, O, that is a - O 0
we hdx X?,b
we have x =- + / -, + =2 l/cb.
r Cl