34 
CALCULUS 
8. Evaluate : 
■Sir- 
dx 
p — a or /?. 
(x — p) V c(x — a)(x — ft) 
9. Give all the details of the proof of Formulas (15), (16), con 
sidering, when necessary, the case that x < 0. 
13. Conclusion ; the Actual Computation. Any rational function 
R(x, y) can be written in the form : 
R(x,y)= gfoy), 
V ' 0(x,y)’ 
where g(x, y) and G(x, y) are polynomials. If 
y = Va + bx -f cx 2 , 
the even powers of y are polynomials in x and can be replaced by 
these values. The odd powers can be written each as the product of 
y by an even power, and the latter factor can be replaced by a poly 
nomial. Thus R reduces to the form : 
R(x, y) = 
A(x)+yB(x) 
C(x) + y D(x) ’ 
where A{x), . . . , D{x) are polynomials. 
The denominator can be rationalized in the usual way by multi 
plying numerator and denominator by C — yD. Thus 
R(x, y)= P (x)+ ya(x), 
where p(x), cr(x) are rational functions. Finally we can write: 
.v)= p(x)+ r(x) ■ 
, y 
where r is rational. 
Turning now to the integration of R: 
:, y) dx = Jo(x) dx + I*r(x) y, 
we have first the integral of a rational function, and the method of 
partial fractions, as above set forth, leads to the desired evaluation. 
In the second integral, let t(x) be expressed in terms of partial 
fractions. We are thus led to integrals of the following types : 
Rx n dx r dy i* 
J y J 0 - p) n y J < 
dx 
O® 2 + px + q) n y S( x2 + P x + Q) n y 
xdx 
These integrals are computed by the aid of Reduction Formulas: 
cf. Chap. II.