12 Prof. J. R. Young on the Integral C — , and on 



X denoted by / — to vanish when x=\^ whilst the equation 



%J X 



/x"dx= — — supposes I x"dx to vanish when x = 0'' It ap- 

 pears to me that this explanation is by no means sufficient to 

 justify the assertion that the general form Jails. Every student 

 ot the calculus knows that, by integrating the same expression 

 by different methods, different functions of the variable will 

 often arise, which can only become identical, in particular 

 applications of the results, when each is connected with its 

 own supplementary constant. One method may lead us to 

 logarithmic functions, another to circular; and though they 

 both arise from one and the same differential, they cannot, in 

 general, be equated till each has received its own peculiar 

 correction. In such instances it appears to me that it would 

 be just as proper to say that one of these methods ya//s, as in 

 the instance before us. The fact is, that in all cases of general 

 integration, where the supplementary constant is suppressed, 

 the process is really performed between limits, one of which 

 is fixed, and the other arbitrary. To be strictly accurate, 



/dx ' /^^ dx 



— = log X should be written / — = log x. 



Introducing this accuracy of expression, let us now return 

 to the general form, which in the case under consideration 



x~^dx; or, for convenience, changing x into l+z, 



x'^dx. Bythe general form, the value of this is- 

 Developing by the binomial theorem, we have 



X 







A Z.o 



and consequendy 



the known development of log (1 +z), or log x; so that the 



general form really gives us / x~^dx= log a*, without any fail- 



ure at all. And we thus get moreover the interesting symbo- 

 lical result, 



