INTEGRATION OF ALGEBRAIC FRACTIONS 151 



(So? 3 - 15) dx 



1 hen 



- 7 



dx 



7x - 



f* /~ I /* 



I / *)np _L_ i\fi'y 



2031- ' - 767 I 



4 la? 2 +7a:-7 2 la: 2 



203 / 7\ 767 . 



- log, (a; 2 + 7a? - - ) - - log, 

 \ T</ 4 v 14 



(d) Care should be taken when the term involving a; 2 in the de- 

 nominator is negative, for then the denominator reduces to the 

 form a(A 2 - X 2 ). 



To integrate 



Now 8 - 12a? - x* = 44 - (a: 2 + 12a; + 36) 

 = 44 - (a: + 6) 2 



(* dx f ckc 



J8- 12a; - a; 2 = J44^ 



Then 



( l r+6) 



A 2 - X 2 



where A 2 = 44 and X = x + 6 



Now J * , P 



A 2 -X 2 ~A+X + A-X 



and a (A -ar) + P(A + X) = 1 



1 XT' A "* * ** " ** 



when X = A 



when X = A 2Aa =1 a = r- 



2A 



fdX irfdX ffiX'l 

 Then J^-^ ^-U^-^ + J^- _| 



,i/f 



2AUA+X A-X 



-!A^ 



1 . A + X 

 = 2A log A^X_ 



1 2VIT+ x+ 6 



= 4VIT oge 2 VIT - x - 6 



In dealing with l-r ^. we notice that the numerator is not 



JA A 



the differential coefficient of the denominator, but making the 

 integral negative and the numerator negative such is the case. 



Thus 



