18 DIFFERENTIAL AND INTEGRAL CALCULUS. 



f<f) (x) dx = \jr (x) + (7, as meaning that the differential co- 

 efficient i|r' (x) of ^ (x) is $ (x) 5 or, in other words, that 

 the integral of </> (a;) is ty (x) + C. Such result is called 

 an indefinite integral. If no other data are given, (7 cannot 

 be determined, but every value will answer the question. 



Thus if -~ = - 7-, we know that -=- (\/Wl = 



dx 2 VH dx p 2 V(^) 



and therefore -7^ = 2 -T- (V#) = -7- {2 V(#)}, and therefore 

 dx dx dx 



-j- (y 2\A) = ; whence we must have y 2 \/(x) = C, 



where C may be any constant quantity. Suppose, however, 

 we have the farther datum that when x= 1, 2/ = 0, then we 

 have 2= (7, i.e. (7 = 2, and the general relation is 

 i/ = 2 + 2 \/(x). Such a result is called a corrected in- 

 tegral. If, finally, the thing sought be the value of y when 

 x = 4, then we get from this equation y = 2. This result 

 is written 



dx 

 -T = 2 and is called a definite integral. 



In general 



I (f> (x) dx = ty (x) + C the indefinite integral. 



r$ (x) dx = i|r (x) "*jr (a) corrected integral. 

 .. 



<!> (x) dx = -Jr (b) ifr (a) definite integral. 

 



The case in which (7=0 is very frequent, and such 

 a result is generally written /$ (x) dx = ty (x) (+ (7= 0). 

 The best illustration of integration is given by considering 

 the result sought for as the area of a curve bounded by 

 the axis of #, any two ordinates, and the curve. Thus 

 suppose aPQb (fig. 4) to be a curve, the ordinate MP at 

 any point being a known function <f>(x] of OM or x the 

 abscissa, take ON= x + Ao?, NQ = y + A?/ = </> (x + A.r), 



