THE INTEGRALS J^== AND IVXMT 2 dX 168 



(b) When the fraction has for a numerator a linear function 

 of x, before proceeding to integration, the fraction must be split 

 up into two fractions, the first of which must have for its 

 numerator the differential coefficient of the quadratic expression 

 under the square root. 



(b) To integrate 



. 

 Vx z - ISx + 106 



- 5 2(2# - 18) 31 



_,_ _._ _ ^ _ ____., _ ___ 



Va? 2 -lSx+ 106 ^x 2 - ISx + 106 \/ 2 - l&e + 106 



f 

 J 



(20 - 18) da? f dy 



, , = ~7= where y = x 2 - ISx + 106 



Vx* - ISx + 106 J Vy 



Also 



= 2Vx z - ISx + 106 

 dec ( dx 



+ 106 JV(#-9) 2 +25 



where X = x 9 and A 2 = 25 



JVX 2 + A 2 



,x rx+ 



smh- 1 -^ or log e ( 



x-9 ((x - 9) + Vx z - ISx + 106 \ 

 orlog^ 1 -| 



Then 



r (4# - 5) dx f (2a?- 18) dr P 



]Vx z - I8x+ 106" JVa: 2 - l&c + 106 J V 2 - 



ISx + 106 



= 4 Vx* - ISx + 106 + 31 sinh- 1 



(x - 9) + Vic 2 - 18 + 106\ 

 or 4 2 - I8x + 106 + 31 log, - ' - 5 - / 



(c) When the square root of the quadratic expression appears 

 in the numerator, the same substitution can be used for X, but 

 a different integral is the result. 



(c) To integrate V# 2 + 24c + 244 

 J Vx* + 24a; + 244 dx = j V(x + 12) 2 + 100 dx 



[ 



VX 2 + A 2 dX where X = x + 12 



and A 2 = 100 

 put X 2 = A 2 sinh 2 



