170 



PRACTICAL MATHEMATICS 



(d) When the numerator of the fraction is of the second degree, 

 the fraction must be split up into three fractions. 



2 2 -7x+IO 



. ,. _ . . 



(d) To integrate 



, 

 Vx 2 -Q-x+6 



Then 



2x 2 - 



x 2 - 6x+ 6 



2(x 2 - 6x + 6) 



5x- 2 



= 2\/x 2 - Qx + 6 2 Vtf 2 - 6x + Q 

 and each term can be integrated separately. 



j Vx 2 -Qx+6dx = \V(x-3) 2 -3 



dx 



- J\/X 2 -A 2 dXwhereX=;r-3 andA 2 =3 



A 2 fXVX 2 - A 2 . X 

 7-5 cosh" 1 



3f(x- 3}Vx 2 - 6x+ 6 



-6)dx [dy 



== ^ == 7 = ^^ 177 s where y = a; 2 - Qx + Q 

 6x + 6 J vy 



f da; j* dx 



J Vx 2 - Qx + 6 = J V(-3) 2 - 



= f 



- 6x+ 6 



dx 



3 



dK 



JVX 2 - A 2 



where X = x - 3 and A 2 = 3 



, , 



= cosh- 1 -T- 

 A 



Then 



f- 



= cosh- 1 



x 2 - 7x + 10) dx 

 Vx 2 - Qx + 6 



x- 3 



f 5f (2aj-6) dr , f dx 



= 2 \Vx 2 - Qx+ 6 cto+ - > + 13 = 



J 2J v 2 -6^+6 J Vx 2 - 6i 



