150 



PRACTICAL MATHEMATICS 



of which can be integrated at once, while the second can be inte- 

 grated by the method of the previous example. 

 dx f dx 



5~ J( 



6) 2 - 21 



dx 



X 2 - A 2 



where X = as + 6 and A 2 = 21 



1 X-A 



~2A gfl X+A 



1 x + 6 - A/21 



2\721 ge x + 6 + \/21 



:n 





12a?+15 



-Slog, 

 2 



(c) To integrate 



- 15 



4a;2+28a; _ 7 



In this example the numerator is of higher degree than the 

 denominator, and the first step then is to divide the numerator 

 by the denominator. 



Then 



8x 3 - 15 



-7 



= 2# - 14 + 



406# - 113 



- 7 

 1 (406a? - 113- 



1 l203(2x + 7) 

 2#-14+-l S 



4i . 7 



1534 



Also 



f 



7 x* + 7ff - 7 1 

 4 4J 



(to 



"-\ 



dx 



rfX 



where X = x + - and A 2 = 14 



A 



1 X-A 



~2A ge X+A 



x+ - 



JL 



1 



2A/14 



_ 



2a? + 7 - 2 Vl4 



2x + 7 



