INTEGRATION OF ALGEBRAIC FRACTIONS 149 



Hence or 2 + bx + c - a(X 2 + A 2 ) or a(X 2 - A 2 ) 



b 4>ac b 2 b 2 4ae 



where X = x + and A 2 = r or - * 

 2a 4a 2 4a 2 



Thus a quadratic expression which will not factor may be 

 expressed as the difference or the sum of two squares. 



Case I. When the denominator of the fraction reduces to the 

 form a(X 2 - A 2 ). 



(a) To integrate - - 



x z + lOx + 13 



Then x 2 + Wx + 13 = x 2 + lOx +25-12 



= (x + 5) 2 - 12 



= X 2 - A 2 



where X = x + 5, A 2 = 12, and also dX = dx. 



Then f dr f ** 



J* 2 +10 t r+13~JX 2 -A 2 



But 



X 2 -A 2 X + A ' X-A 

 and a(X - A) + p(X + A) = 1 



whenX=A 2AB = 1 ft = -L 



2A 



when X = A 2Aa =1 a = r 



2A 



Thus ' 1 



f dX 

 JX 2 -A 2 ~ 



= ^.{lo g< (X-A)-log e (X+A)} 



1 X-A 



~2A 10ge X+A 



f dx I x + 5 - 2 \/3 



and - = r= log.. - r=. 



}x 2 + lOx + 13 4A/3 x + 5 + 2 A/3 



(6) To integrate 

 Then 



x z + I2x + 15 

 5x - 4 5 2a; + 12 34 



x 2 + I2x +15 2x 2 + I2x +15 # 2 + I2x + 15 

 The first fraction is one obtained by making the numerator 



the differential coefficient of the denominator ; the multiplier - 



is so chosen that full account is taken of the part 5x in the 

 numerator of the original fraction. This has the effect of re- 

 placing the fraction to be integrated by two fractions, the first 



