148 PRACTICAL MATHEMATICS 



Cdt/ dti \ dii r 



putting y = x + 2, becomes I -f-, since -p = 1 and I -Y = I 2/~ 2 



Cose IV. When the numerator is of higher degree than the 

 denominator. 



x 5 2 + x - 2x 2 



By division 7-3 -- - -r = x 2 - x + 2 + 7 -^ 

 2 z 



2 + x - 2x* A B 



(a; 2 !)(#+ 1) +l (a? + l) 2 # 1 

 and A(a? +!)( 1) + B(a? - 1) + C(aj + I) 2 = 2 + a? - 2# 2 



when a?=l 4C=1 C=- 



4 



when =-! -2B =_iB = | 



when x= -A-B+C =2 A - - ? 



4 



a; 5 due 

 Hence 



;__ 1 f dx 1 f rfa; 



T + 2 ](x+ I) 2 + 4 J t r~ : T 



12) - ^ - (9 log e (ar+ 1) - log, (*-!)} 



89. We next have to consider the integration of algebraic 

 fractions the denominators of which are of the second degree but 

 cannot be resolved into linear factors. In this case the method 

 of integration depends upon the nature of the denominator. 



For ox 2 + bx + c = a ( x z + - x + - } 

 \ a a) 



a 4a 2 / \a 4a 2 /J 

 & \ 2 /4ac & 2 \1 



b \ 2 



according as 4ac is greater or less than b 2 . 



