THE INTEGRALS [ ' " v . AND f\/A 2 -X 2 dX 159 

 JvA 2 X- J 



Jso I . 



sin- 1 - = 

 VlQ9 



sm 



\/109 



/fit* ^ Q\ /7'i /* / O/rt T\ 



^ Otc/ 5? I f *\ * I I &JC ~~ < I 



I = 4 i- 



JVl5-7tf-tf 2 J 



\/l 'i 7-r T^ 

 V A3 <!* it* 



-37 



J \/15 - 7a; - a; 2 



, - 

 = - 8\/15 - 7a? - x 2 - 37 sm- 1 -7= 



(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 \/32 + 18* - x 2 . 



= j V32 + I8x - x i dx = I \/113 -(as- 9) 2 dx 



put 

 Then 



= \/A 2 - X 2 dX where X = * - 9 

 and A 2 =113 



X 2 =A 2 sin 2 6 

 A 2 - X 2 = A Vl - sin 2 = A cos 6 



jn 



also X = A sin 6, JTT = A cos 0, and eZX = A cos dQ 

 aX 



Then 



J\/A 2 - 



X 2 



= A 2 [cos 2 d0 

 = ^- f (1 + cos 20) d0 

 = ^{0+1 sin 20} 



,X 



A+ 



