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.

208

The integral of explicitly irrational functions. Bk. III. ch. III.

we have:
Jdx Jx cl xd -Xi x1d X dx _ cxdxl - xdx3
y 3Z2 XJ (X-Q)y J (X1- X3) 2
Hence we see, what is lost sight of in the non-homogeneous form,
that the first integral is only a special form of the second, since
it involves the intersections of the conic with the right line x3 = 0,
instead of those with the right line x - QX3 0. = (Aronhold loc. cit.).
When f() = 0, the right line x = Q is a tangent to the curve,
for, its two intersections have the same coordinates x = Q, y= 0;
but then:
__dx 2 1V X
(x - Q) VR~ - f () x - Q
and for x= 9 this algebraic expression becomes:
_ C 2 b+cx) = ( 1/f) o
f\CQ) X- e f (Q) VR JB 1 f(Q)
3=Q X=Q
The same holds for the parabola where it meets the line at infinity.
123. From the indefinite integral we can form the definite for
two limits between which the integral function remains continuous
and does not become infinite; thus for instance, if 92 > 1, we have by
formula III' ~ 122:
J^ d()F ~, \ ]/^ tan- +//i '- )_
+1 __ X_ — _
f(XQ1 _ - ( e a1 J _-x
When x -- 1 the argument of this circular function vanishes; its
values are continuous and negative as x increases, when x =- - 1 the
value becomes -oo; for, we have
x - x 1 _ -/l - x V -_ / V/ 1S _1
the factor / Q 1 is positive, therefore we have:
+1
-1
This reasoning would not be possible if Q2 < 1.; in this case tan-~ is
discontinuous in the interval; the value of its argument is - i
when x Q.
124. The class of integrals of explicitly irrational functions next
in order is given by JF(x, 1/R) dx; R being a polynomial of the third
or fourth degree in x,
- == a + bx 4- cx2 + dx3 + ex,4