THE INTEGRALS | yA 7~ V a AND fVA'-X 2 dX 161 



'* J 



Ida; 



V06 - (a; +7) 2 



where X =a;+7 and A 2 =66 



f ^ f 



J\/17- l*x- x 2 J 



VA 2 - X 2 



i x 

 sin" 1 -r 



A 



V66 



P 2a; 2 - 8a; + 9 f . 



Finally <fo = - 2 U/17 - 



J V17 14a; a; 2 J 





Vl7 14j; a; 2 JV17 14a; a; 2 



a; + 7 , , 



- 66 sin- 1 77^1 - (x + 7)Vl7 - 14a? -a; 2 + 36V17 - 14a? - 



+ 295 sin- 1 



V66 



^^ _ . x I 7 



= (29 - )Vl7 - 14# - a; 2 + 229 sin- 1 j= 



It is evident that integrals of this type depend upon two 

 standard forms. 



-^ r 



95. Cfwe //. (a) When the denominator of the fraction reduces 

 to the form VX 2 + A 2 , for this type we have to use hyperbolic 

 functions. 



(a) To integrate . 



Va; 2 + I2x + 48 



Now a; 2 + 12a; +48 = a: 2 + 12a; + 36 + 12 



= (x + 6) 2 + 12 

 dx 



Then 



-f; 



+ 48 J V(x + 6) 2 + 12 



where X = x + 6 and A 2 - 12 



put 



