605 



INTEGRALS, DEFINITE. 



INTEGRATION. 



906 



The process in the article AREA will now easily show that, y being 

 the ordinate of a curve to the abscissa x, the area contained between 

 the ordinates whose abscissa; are a and a + b, the part of the abscissa 6, 

 and the curve, is fydx taken from x=a to x=a + t. Thus if the curve 

 be a part of a rectangular hyperbola, whose equation is xy=c, or y= 



, the area included between the ordinates, whose abscissa; are 1 and 

 x 



1 + k, is fdx from x=l to x=l+k. But c log a; is the function 



whose differential coefficient is _- ; whence it follows that the pre- 



x 



ceding area is c log (1 + 1) c log 1 or c log (1 + 1) square units. This 

 is the property of the hyperbola from which the logarithms of Napier 

 were called hyperbolic. [LOGARITHMS.] 



An integral is said to be definite, when its limits are given ; and in- 

 definite when they are not given. 



INTEGRALS, DEFINITE. [INTEGRATION, DEFINITE.] 



INTEGRATION. In the article INTEGRAL CALCTTLUS, the meaning 

 of an integral was explained. The present article is devoted to the 

 operation of integration, that is, of finding the primitive function 

 which has a given function for its differential coefficient. Having given 

 p a function of x required q so that d<t : dx may be r. In the article 

 QUADRATURES, METHOD OF, is given the mode to which we must 

 have recourse, in order to find particular values of Q, when the 

 general methods for determining it fa.l. In this article we confine 

 ourselves to what is most useful in operation, as a summary for the 

 advanced student, not an explanation for the learner. Properly speak- 

 ing, the problem requires some addition to make it definite. Thus -.< 

 has x' for a primitive function, and also x* + c, c being any constant 

 quantity whatever. In the present article, we shall neglect this con- 

 stant altogether, reminding the reader that he must never omit it in 

 any application. If he should find in different books different functions 

 given as the primitives of one and the same function, he will always 

 find that those different primitives differ only by a constant quantity. 

 Thus (1 x)~ ' and a(l -a:)~ l both occur as the primitive of (1 a,')~ a i 

 but they only differ by a constant, namely 1. 



In the common process of integration, the actual passage from the 

 differential coefficient to the primitive is always an act of memory. 

 The algebraical work which occurs is always used either to reduce a 

 form in which memory will not serve into one in which it will, or else 

 to reduce the given differential coefficient to two terms, one of which 

 can be integrated by memory, and the other of which is more simple 

 than the original quantity. 



The functions in which the simple remembrance of the forms of the 

 differential calculus is of use are as follow! : 



dx 



dx 



dx I 



? - (T 





dx 



"dx - 



/cos*<ir=sin*,./*c^ =tana: 

 -dx 



dx 



To these should be added the following, which may be obtained in 

 various ways from the methods of this article, or from peculiar artifices 

 which are found in works on the subject. 



/* <** 1 . / + *\ r dx I (x-a 



' J a'-x> ~ 2a lo \^x) J *=* = To, Io 8 \7^ 



dx 



= a ten -a 



dx 



/dx x f dx (* x\ 



STZ = 1 8 tan 2y c^ = 1 K cot U - 2] 



Jteaxdx -log cos x J' cot x dx log sin x. 



Among the peculiar artifices of integration may be reckoned the 

 following, which are perhaps nearly all that can be useful to a 

 learner : 



1. The reduction of inch a form as f\dx to another form f\dv, in 

 which r i a different variable. Thus /(<j j + 3?)'xdx can be imm. 



diately reduced to \J(a- + x 2 ) " d. (a? + a?) or ^fv'dv, where means 

 a- + i-'. The second form is immediately seen to be integrable. Cases 

 of this kind are so various that the student must form the habit o 

 looking for them, and recognising them at sight. Sometimes a slight 

 transformation is required, thus : (1 + at*)~ l dx, when reduced to 

 (f~* + a)~ 1 f~*dx clearly shows the form J~'rfr, where v is 

 -* + <*. 



2. The reduction of algebraical to trigonometrical functions, and 

 the converse. Thus (a 5 x')"x*dx, if x be made a sin d, becomes 

 a tmr*-n C o8+i9 shvflrffl. Also/ (sin 9, cos 8) . de,ii a;=sin 0, becomes 



f\x, ^(1-*S)}.(1 -<*)-<**. 



3. When rational powers appear in a denominator, they should be 

 transferred to the numerator by changing a: into 1 : z. By such a 

 transformation, we change 



dx z-Vz 



into - 



4. When an irrational root of a polynomial appears in the nume- 

 rator, it should generally be transferred to the denominator : thus, 

 Vx rf.t should be written xcte : Vx. By such a transformation, we 

 change 



V (' + #*) dx into 



5. When, by the addition of more simple terms to the numerator, it 

 can be made the differential of the prominent function of the denomi- 

 nator, such additions, with compensating subtractions, will frequently 

 reduce the question of integration to a more simple one. Thus we 

 alter 



xdx _ 2_ 



" * 2c 



2e 



2c V(a + bx + ex*) 



the first term of which can be integrated as in (1), leaving the second 

 term, which can be simply integrated. 



6. The process known by the name of integration by parti, consists in 

 reducing the form ndx into any convenient form vrfp, and using the 

 obvious theorem 



fvdv - vi> fvdv, 



thus the finding ot/\dv is reduced to that of /i*rfv, which it may often 

 happen is the more simple of the two. Thus to find/c" log xd.c, we 

 have 



/log xd 



about the second term of which there is no difficulty. But it often 

 happens that this method succeeds by a succession of reductions. 

 Thus it gives 



ft'x'dx = x't" 6fe*x"dx 



in which the second term must" be again treated in the same manner; 

 and so on, until we arrive at ft * dx. 



7. In the last mode of proceeding, it is best to form, in general 

 terms, an equation of reduction, as it may be called, which furnishes 

 the key to the reduction of each case to the one below it. Thus if 

 ft"x*dx be considered as a function of , and called v, integration 

 by parts gives 



thus showing how to find ft "x * dx aa soon as ft x*-^dx is known. 



8. The use of the equation of reduction depends upon our being 

 able at last to reduce the question to that of finding a visibly known 

 integral. Thus, if in the preceding n be an integer, we must at last 

 come to ff"xdx, or ft "dx, which is known. But if u were a fraction, 

 no reduction of the value of n by units at a time would lead to an 

 iutegrable form. 



9. The integrable form at which we arrive by successive reductions 

 is called the ultimate form. It frequently happens however that the 

 reductions proceed by two or more steps at a time, in which case two 

 or more ultimate forms result. For instance v. =/(a 2 x 2 )l x *dx 

 has for its equation of reduction 



x* 1 V(<** *) n 1 



Accordingly, when n is even, we are brought at last to V,,, and when 

 n is odd, to V,, or to sin - >(x : a) and ^(o 2 a*). 



10. In using equations of reduction, it will be found more conve- 

 nient to work upwards from the ultimate form to the case required, 

 than in the contrary way. Thus if we want v 4 =/f *x*dx, the equation 

 of reduction being 



v. =*" nv,-!, 

 we should proceed as follows : 

 V =", v l= xt* -' 



