158 PRACTICAL MATHEMATICS 



Case I. (a) When the denominator reduces to the form 

 A/ A 2 X 2 . This is obviously the case when the term involving 

 x 2 in the quadratic expression is negative. 



(a) To integrate .. 



A/8 - 120 - x* 



Then 8 - I2x - x 2 = 44 - (x z + I2x + 36) 



= 44 - (x + 6) 2 



f dx f dx 



Hence I . = I , 



J A/8 - I2x -x 2 J A/44 -(x+ 6) 2 



f dX 

 = 1 . - = where X= x + 6 and A 2 =44 



JA/A 2 -X 2 



put X 2 = A 2 sin 2 



Then VA 2 - X 2 = A A/1 - sin 2 = A cos 



,7V" 



and X = A sin 0, -^ = A cos 0, and dX = A cos d0 

 at) 



f < f 



Therefore JvA 2 -X 2 = J 



A cos 



A cos 



= 



i x 

 = sm" 1 -r- 



A 



. . x+ 6 

 sin" 1 =. 



f 



and finally, . 



J-V/8- 



(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. 



Sx 9 

 (b) To integrate 



Then 8X ~ Q 



A/15 - 7x - x z 



f ( 2x 7} dx f du 



Now 1 v , ' = I j= where y = 15 7x + <u 2 



jVl5-7x-x z JV 



15 - 7x - x 2 A/15 - 7x - x 2 



= 2 A/15 - 7x - x 2 



