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.

190 The integral of rational algebraic functions. Bk. III. ch. II.
PaQ- dz P_, - a+ x-a
J z ^+ I - tan-' -
Accordingly:
IL X a2,+ 3 2 d x =- ax -+-a q2+ Q tan-' cx
Treated thus, the last Example gives the value:
f- j dx - l(x2 + 1) + tan-x + C.
Moreover the definite integral can now be calculated at once from
this for any finite real interval by the formula:
b
=X + 1 1
j d1 x= l A — +- tan-1 b - tan-1 a.
From the two different forms found for the same integral it is
obvious that we must have for all values of x,
tan-' =x - (2 ) + Const..
But this is in fact involved in ~ 74 which gives:
x- i x2 - 1 - 2xi 2x
x-i =1 X - - itan — — + C == 2i tan-'x + C.
Le + a 2 4- -
Example:
dx dx 1 cx-t-b
ijo b x =cx fc +b)+ac —' Iacf 1 tan-1 c- - c.
J b + 2 = == C J (CX j- 2)2+ac — b2 == y-2 n ac-b'
This expression is real, provided ac - b2 > 0, i. e. when the roots are
complex. Under the same condition we have the definite integral:
A_ -r d — = - c, otherwise this formula does not hold.
j a-+ bx-+ cx2 V/ac - b2
113. If the function (p have multiple roots:
( (x) - (x a)(x- a) - (x - (x- ),2, 1 + A2 + + - n,
the foregoing process of resolution into partial fractions no longer
applies; but we have now the theorem: The quotient (x) can be
always resolved in one way only into the form:
ip (X ) A, _ _ _ _ _ _ _ _ _ _ _
1 () -(x) = - +( - A )21() -, where qp(x)=(x-ca)2l p(x),
A, denoting a constant and l, (x) an integer function of the order
n - 2 at most. This follows from the identity:
2) l(x) = Aop, (x) + (x- a),(x) or:,I(x) = (x) Ao,(
X - c