INTEGRATION. 



INTEGRATION. 



(08 



11. There are several cases in which th following extension of the 

 theorem known by the name of John Bernoulli may be useful. Let 

 ', T, Ac. be the successive differential coefficient* of * with respect to 

 x, and let r,, r,, r,, 4c. be the successive integrals of r with renpect to 

 jr: then 



This U particularly useful when M is a rational and integral function, 

 and is successively intcgrable with ease, ae when u is t *, ain ax, or coa 

 x. The process can then be continued until the remainder vanishes. 



12. In the case of Qxdjc : $jr, where fur and tyx are rational and 

 integral functions, the integration is always possible so noon as all the 

 roots of tr=0 are found. The process in FRACTIOUS, DECOMPOSITION 

 OP, must be applied. When this is done, and the function thereby 

 reduced to the sum of terms of the form A(X a)~*rf.r, the integration 

 gives no trouble. 



13. In the case of a pair of irrational roots, o+S v 1, each occur- 

 ring once, the sum of the terms which they produce can be reduced to 

 MM form 



the integral of which is 







14. When junlxit a function of powers of any one case of ar + 6, it 

 can, if irrational, be reduced to a rational function by assuming 

 <vr + 6 = r", where m is the least common multiple of all the denomi- 

 nators in the exponents. For dx becomes m*~ l rff : o, and every 

 power of ox -I- 6 becomes an integer power of r. 



15. The function x" (ax + 6) dx can be integrated when either m or 

 n is a positive integer : when n is integer, by simple expansion ; when 

 m is integer, but not n, by making ax + 6 = v, and substituting. But 

 when both m and n are negative integers, let x = l : y and after sub- 

 stitution, make a + 6y = ir, and substitute for y. 



16. The function $xdjc ; (x* -f-o * ) can be easily integrated by de- 

 composition of fractions, the denominator never having equal roots. 

 The same may be said if we substitute j- s " + 26a " .r " + a* in the 

 denominator. 



17. In x'(a + bx' )'dx we have an integrable function, whenever 

 either of the following is a positive integer ; 



r+1 r+1 



,or- _, 



The substitutions which succeed in the two cases are 

 a + bx* = e 8 , and <- + 6= 8 



8 being the denominator of (. 



1 8. The following transformation involves a large number of obvious 

 cases, and is constantly occurring. If fiptdx = ^a:, then f<t(ax + b)dx = 



Thus in no list would /cos(aar + b)dx be set down, after /cosa; dx 

 has been given. 



19. The following integrals are worth giving separately as ultimate 

 forms ; 



dx 1 a 



-TV = - cos-' -, 



/ 

 / 

 / 

 / 



/; 



dx 



*V(a':r) 

 dx 



=coir ~ l "~ir =venr ' 1 1 



rf(2et + ft) 



which come* under one or another of three previously given forms 

 according as & toe is positive, nothing, or negative. 



/ 



xdx 



- f 



2c J a 



dx 



f 



2cx-b 



V(o + 6. c 

 V ; + * dx - -s : 



V 



. , 

 """' 



Y tog (JT + 



bx 



)\ \ 



x 1 + 



I-et x = (a -t-ftz !(*). Thra 



* , 

 Vx . <i - 



f 



J Vx 



/jAe Vx J>_ 

 vx = ~c~ '~ 27 



dx 



.hr* 



cor 



6+a coax 



a + bcosx V(o"-**) "* " o+o cos z 

 1 _ ( 6+0 cos a+ VXEi'-a^.sin x\ 

 o+ocosz J 



20. All that it remains to give are the equations of reduction for 

 remarkable cases. Many other differentials can be integrated in a 

 finite form : but it is impossible to give a list of all which ar some- 

 times useful. The transformation of unknown to known forms U one 

 of the most necessary studies of the young mathematician. 



21. "Let \n, . =/(log i)"a* dx. Then 



*+' 



22. Let p stand for A2 + at*, (m, n) for fx*r*dx, g and h for 

 + ! + < and m. + 1 + nb, and e'for a b. We shall then have 



A(t, n) + NA(m + a, 1)= *"*> i-" 

 0(ro, ) ncB(m + 5, n l)=a^"+'F" 

 <7A(m, n) + (A-c)fl(m - c, n) =+' r"+> 

 to(m, n) 



from the first pair of which formula) of reduction can be found for n, 

 be it positive or negative, and for m from the second pair. The 

 most useful cases are those in which a = 0, 6 = 1, and in which 

 a^O, 6 = 2. 



23. Let v. =y\a~o j )-(te. Then 



)-"- 1 ' -8 



24. Let v. =J(aff)*dx. Then 



25. Let v=/(a s x*)-*dx or J\a?-x>)'dx. The equations of 

 reduction are those in (23) (using the + in ) and in (24), and 

 writing d'x 1 for a^a 2 . 



26. Letv,.=/r"(a ! * s )(te. Then 



1 -! 



- 2<-l)a '"- 

 _ m-1 



rr*,<*'v-*. 



27. Let v 



Then 



28. Instead of giving a large number of forms which are all de- 

 rivable from (22), it will be better to give an instance of the derivation 

 in full. Let the case be /r-(2oz j^'d*, and let the formula be 

 required to reduce both m and n in numerical magnitude. Here, to 

 transform the formula in (22), 



For m write m j retain n. 



For A write 2a ; for B, 1. 



For a write 1 ; for 6, 2. 



For 17, m + l + ; for k, m+l-r2/i. 



For c write 1. 



The first formula connects ( m, n) and ( m 1, 1), the second 

 ( m, n) and (--2, n l)j_the third ( m, n) and ( m 1, n); 

 the fourth (- m, n) and ( m + 1, n). By either of the first two we can 

 therefore reduce both ; by either of the last two we can reduce m only. 

 Observe that whenever a formula will serve to raise either exponent 

 it will also serve to reduce it. Thus, if a formula ware 



