DIFFERENTIAL AND INTEGRAL CALCULUS. 17 



INTEGRAL CALCULUS. FIRST PRINCIPLES. 



Having investigated methods of finding the differential 

 coefficients of any functions of a?, the next thing in the 

 natural order of the subject (just as division succeeds 

 multiplication) is to reverse the process, and to enquire 

 after methods which shall enable us, when the differential 

 coefficient is given, to determine the quantity of which it 

 is the differential coefficient. This is a very different 

 matter, and we cannot assert that it is always possible. 

 If a function can be expressed in terms of x by any known 

 algebraical symbols we can find its differential coefficient, 

 but we can only reverse all these processes to furnish us 

 with integrals of certain functions of #, and if we cannot 

 by any of our methods reduce a quantity whose integral 

 is sought to coincidence with a known result of differen- 

 tiation, we are at a stand. 



Thus having investigated the properties of the function 

 of x which we call its logarithm, and having found that 



-7- (log x) is-, we say (meaning the same thing, neither 

 dx x 



more nor less) that the integral of - is logo; ; but if we 



x 



had not known the properties of this function we could not 



have integrated - . No one can integrate 4 and 



x V I-'- x ) 



many other apparently simple functions. 



The notation adopted in the Integral Calculus is as 



follows : if -j- = $ (a?), and y = "fy(x), then f<f> (x) dx ^r (x) , 



or more correctly = ty (x) 4- (7, where C may be any constant 

 quantity that is, any quantity independent of x. This 



constant is necessary since -=- [ty (x)} = -j- ty (x] + C}. 



The reason of the notation will be explained subsequently ; 

 at present it will be better simply to accept the equation 



D 



