SUPPLEMENT TO CHAP. VU. [CHAP. TL 



/2 of r n j-"' r 1 



2xdx=~=x* J3x*dx = ~ = e> l 



/*, o>* 2 J / na- 



r rf 3= _ = -x I n x*-i dx = - = a?>, 

 f 8 J n 



etc. 



It is of this latter rule that we shall make especial use in what follows. 

 0. Other Principles Integration between limits, etc. 



We may observe from (1) and (4) that a constant factor may ~be put out- 



tide tJie sign of integration. Thus I 5x2xdx=5 I2xdx = 5x > . 



It is also evident without demonstration that the integral of the sum of 

 any number of differential expression* is equal to the sum of the several 

 integrals. 



Thus / Fa; cf + * # e + y* dy + x* dx\ 



is the same as / x dx + I &dz+ I y* dy, eta 



If in (1) we had 



y = 5 *+ 

 where a is a constant, we should have 



y+d y = 5 (x+dx)* + a=:5 (** + 2 xd x+dx*)+a, 



or dy = 5(2xdx + dx t ), or ^- = 5 (2x + d x): 



a x 



whence 



- = 5 x 2 a?, or d y = 5 x 2 x d x, or 



il x 



just the same as before. 



The integral of this will then be y = 5 x* as before, whereas it should be 

 y = 5 * -f- a. 



If two differential equations, then, are equal, it does not necessarily follow 

 that the quantities from which they were derived are equal. 



We should, then, never forget when we integrate to annex a constant. The 

 value of this constant will in any given case be determined by the limits 

 between which the integration is to be performed. 



We indicate these limits by placing them above and below the integral 

 sign. Thus the integral oix*dx between the limits of x = + h and *= h is 



r +h r # 



I x* dx. If we integrate a* d x, we have, then, I x* dx = + O, 

 J h J 



where O is a constant whose value must be determined by the conditions 

 of the special case considered. If we introduce the value of x = h for one 



limit, we have + O. For * = 2 h for another limit, we have - + O, 

 3 o 



We have, then, two equations, viz. : 

 when x h, 



f h* 



I a? dx = 



Jx = h 



