160 PRACTICAL MATHEMATICS 



Finally, 



, , 113 f . ,0-9 (0-9)\/32 + 180 -0 2 1 



I w >; svi /inri - _ - _ 7 ci n "~J- - - I ' ' I 



A Oti/ / Hi*/ ^T. \ ijliL ' ' ' "" / 



2 \ yil3 113 



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

 before proceeding to integration the fraction must be split up 

 into three fractions, the first of which can be obtained by division. 



20 2 - 80 + 9 

 (d) To integrate 



V 17 - 140 - a:* 

 9 



Vl7 - 140 - x 2 

 - 2(17 - 140 -x 2 ) 360 - 43 



A/17 - 140 - x 2 V 17 - 140 - x 2 



-is(-2x- 14) 295 



. - 

 - 2 Vl7 - 14* - x 2 - 



. - 18( - 2x - 14) 295 



- 2VI ^ ' r Vl7-I*x-x 2+ ^ 



and each of these expressions can be integrated 

 I Vl7 - 14 - x 2 dx = J V66 - (x+ 7) 2 da; 



= f \/A 2 -X 2 <i whereX =a? +7 and A 2 =66 



cos 2 d6 where X 2 = A 2 sin 2 



A 2 1 



-A.J. 



y f(l + COS 20) 



A 2 / . . X XVA 2 - X 2 \ 

 = -- Sln A + - A^ - / 



+ 7 i 



111-1 77^= +2 (*+ 7 )\/17 



f(-2E-U)to fj^ 



J \/17 - 140 - a? 2 JV?/ 



= 2 A/17 - 140 - 



