144 PRACTICAL MATHEMATICS 



dv 1 



and conversely I = log e x 



j x 



This result can be used in a more general sense, for if we con- 



. f (Vox + b) dx 



sider the integral I 5 H as an example on the use of it, 

 J ax 2 + bx + c 



by putting y = ax z + bx + c 



then 



ax 

 and dy can replace (2ax + b) dx in the integral. 



The integral then becomes I = log e y 



j y 



= log e (ax z + bx+ c) 



I b 



It should be noticed that the fraction 5 -. - belongs to a 



ax* + bx + c 



particular type in which the numerator is the differential co- 

 efficient of the denominator, and the above method of treatment 

 will do for all fractions belonging to this class. In general, if 

 we integrate a fraction whose numerator is the differential co- 

 efficient of the denominator, the result will be the Napierian 

 logarithm of the denominator. 



We can now use as standard integrals 



x n dx = - - ....... (1) 



n+ 1 



dx 



(2) 



Cdx 

 J = 



, fdiff. coeff. of denominator 

 and J -- denominator - -^(denominator). . . (3) 



and employ them to integrate expressions which resemble them, 

 or expressions which can ultimately be reduced down to resemble 

 them. 



87. The fraction whose numerator is the differential coefficient 

 of the denominator or of some part of the denominator, can be 

 readily integrated. The following examples will illustrate this 



(a) f cot dV = K 



sin 

 = log e sin 



