BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF M^nvQ W. Sage 1891 A.-AA.'f:?!/.: The original of tliis book is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001533334 ELEMENTARY INTEGRALS A SHORT TABLE COMPILED BY ,( , T. J. I'a. BROMWIGH, Sc.D., F.R.S. Fellow and Lecturer of St. John's College, Cambridge ; University Lecturer in Mathematics. BOWES AND BOWES 1911 s fN.Z-b^S^i INTRODUCTION. 'PHE followiug table of integrals has been drawn up with the hope of lessening the labour often involved in the integration of elementary functions. The list includes all the ordinary standard integrals and formulae of reduction, arranged compactly, so as to allow of ensy reference. Although the tables occupy less space than some which are in existence, yet it is hoped that the present set may prove no less useful in practical work ; for example, it proved possible in many instances to condense three or more separate formulfe into a single reduction formula ; and it is hardly necessary to point out that a given result can be found more quickly in a short than in a long table. A few other useful forms, such as (57), (58), have been added, which do not appear in the ordinary text- books : and in selecting definite integrals, preference has been given to examples which occur in potential-theory and other branches of Applied Mathematics. Some numerical examples have been added to illustrate the general formulae, but it is not intended that these should be regarded in any way as replacing the sets of examples provided by books on the Calculus. Methods of approximate integration — by Simpson's formulae and the planimeter— are briefly formulated in ±he last two sections. TABLE OF CONTENTS. PAGE I. 1 — 9. Fundamental Algebraic Integrals . 7 II. 10—15. Trigonometrical Integrals . . 9 III. 16—19. Integration by Parts ... 10 IV. 20 — 27. Integration by Snbstitntion . 11 V. 28 — 34. Rational Algebraic Integrals . . 13 VI. 35 — 48. Irrational Algebraic Integrals . 15 VII. 49 — 55. Trigonometrical Integrals . . 22 VIII. 56—79. Definite Integrals ... 26 IX. 80—85. Integration of Series ... 32 X. 86—88. Differentiation of Integrals . 34 XI. 89 — 95. Simpson's Formulfe ... 35 XII. 96—100. Planimetric Formula . . 37 ELEMENTARY INTEGRALS. I. Fundamental Algebraic Integuals. ■ 1. \x''dx = ; if w + 1 is not zero. 2. \~ dx = \ogx or log(-a;), according to the sign of w. 3. [e'^dz = -€'"'. J a dx 4. f^, = itan-p). ]d'-\-x^ a \aj i\ f dx _], /x — a\ 1, fa—x = --cotlrM'-) or -itanlr'f-) a \al a \aj or -; — = — =sin'M-) if a>0, f— = tan-'(-j if y=a^-^^ "• -77-3 ^ = sinlr'(-) if a>0. 8- -77-5 5^ = cosh"'(-) if x>a>0. Both 7 and 8 are included in the one general formula 9. j — = log (a? + «/) if y^=x^+c, which follows at once from the equations , , dx dy dx-\-dy X dx=y dy, or — = -^ = ~ ; but the result can also be derived from (7) and (8) by the method given in the Notes below. Notes. 2 and 5. It is often felt to be strange than we can write f dx according as a; — a is positive or negative. A full explanation involves a little acquaintance with the theory of logarithms of complex numbers, from which it appears that log (a; — a) - log {a — x) is independent of x, and is an odd multiple of ttl ; but the beginner should convince himself by differentiation that the integral is really +Iog(a — :r) and not —log (a — a?). 1, 4, and 6. The inverse circular functions sin"'a7, tan"'ar are to be understood as angles between — ^tt and + ^tt ; with this convention they are uniquely determined by x. The square-roots are to be understood as positive num- bers, here and elsewhere, so that P must be positive if we use such an equation as P >^/Q = i\/{P''Q). 5, 7, and 8. The inverse hyperbolic function sinh"'a: is uniquely determined by x; and so is the function tanh~'a;, so long as x is between —1 and +1; while coth~'a; is uniquely determined so long as x does not lie between — 1, +1. But cosh"' a? has two values (x being greater than 1), and of these it is meant that the positive one is to be taken. The beginner may find it instructive to draw rough graphs of the inverse functions. 5 and 9. It is to be remembered that the inverse hyper- bolic functions are in reality logarithms ; and that we can always avoid their use by simple transformations. Thus, if we write s = sinh u, c — cosh u, we find s + c = e" or u = log (s + c). Thus sinh"'5 = log(5 + e) = log|s + V(l +s')l. and cosh"' c = log (s + c) = log \c + '^{c^— 1) j. 9 Similarly, if ^ = tauliM, we find (1 + 0/(1-0=^'" or taiilr'i! = ^log|(l +0/(1-01) -1<^< + 1; and, if 7 = cotliM, we get (7 + l)/(7-l)=e=», and so coth''7 = -|logj(7H- 1)/(7- 1)1, where either 7 > 1 or 7<— 1. II. TllIGONOMETRICAL INTEGRALS. 10. Jsin a; afar = — COS it;, Jcos a: ofa; = + sin a;, 11. Jtana;rfa; = — logcosa;, |cota;rfa; = + logsinar, 12. Jsec.'cofa; = log(seca; + tana;) = siuh~'(tana;) Jcosec xdx = log ( tan \x^ = — log (cosec x + cot a;), 13. Jsin"a:a?a; = ^(a; — sina?cosa;), |cos'''a; fl?a; = ^ (a; + siu a; cos x), 14. Jtan'a; dx — tan x — x, jeofa; dx = — x — cot x, 15. Jsec'^ar afa; = tan a:, Jcosec'''ara?j; = — cota?. Note. The integrals above are suggested at once from the ordinary rules of the differential calculus, with the ex- ception of (12) ; in this case we may introduce the new variable ^ = tan^a; (in agreement with Vll. below). Then we find u = \6&cxdx=\,^--^, = \og[~-\ , and v = l cosec:r dx= rkr = ^^a^- The integrals can also be fonnd by using tanx as a new variable: and the reader may find it instructive to obtain the same results by this method. The former integral [u) has some interesting applications, and it may be convenient to note a few of its transform- 10 ations here ; for simplicity, we suppose the angle x to lie between — \ir and + ^tt, so that t lies between — 1 and + 1 . Then (5) gives at once M = 2 tanlr' t or tanh \u = tan \x ; and so sinhM = tan:r, cosh m = sec x, tanhM = sinx, leading to e''=sec« + tan.r, e"^ = sec x — tana-, as given in the table above. Cayley proposed to write x = gd m, naming the function after Gndermann, who made tables for its use : but this notation is now seldom adopted. However, the function u is in constant practical use by navigators and others who work with Mercator's charts : for this reason its value (multiplied by 10800 /ir, the number of minutes in the radian) is given in all sets of nautical tables and some others (such as Chambers's) under the title. " Meridional Parts." III. iNTJiGEATION BY PAETS. 16. The fundamental formula is \u dv = uv — \v du. 17. If X = le'^cosbx dx, Y= le'^sinbx dx, prove that aX-bY=e'^cosbx, bX + aY=e''^smbx. Deduce the values of X, Y, and so show that X+i,Y = e'^lc if c = a + tb. This result indicates that formula (3) remains true for complex values of the index a. 18. Prove that jar''-Mog^rfx = "^ [\Qgx-^, Ixe'dx = (x-l) e% JxVrf.c = {x'— 2x + 2) e"', ^xVdx=[x''-7ix''-'+n(n-l)x"-'-...{-iynl]e''. 19. Find ^(s[o:^x)dx, ^x(sin'^x)dx, ^xsiaxdx, J(tan"'x) dx, Ja- (tan"' a.-) dx. 11 IV. Integration by substitution. Integrals contaiaiug the general quadratic ^ = az'+2bx-\- c are often simplified by using the substitution v = ax + b, for then aX = v' + I), if D=-ac-b\ 20. Thus, if i* is positive, we find «?a;/Z=(^«j/(t)' 4- D), or . - rdx 1 , _Jax-\-b\ which leads to a convenient formula (57) for definite integrals. When D is negative, the roots of X=Q are real, say a, /3 and we get the formula which reduces (for a definite integral) to a form similar to the first integral. 21. Similarly, if y^=a.r'+ 2bx-\-c, we find d.r^ dv y ~ s/a ^/(v' + U) when a is positive. Hence, from (9), {a) \^ = -j-log{y \/a + ax + b). When a is negative and equal to — oSp we write v^ = a^x — b, a^y''=D^ — 'o^, where D= — D is now necessarily positive. Then, from (6), we find The apparent difference between the two types disappears when definite integrals are considered as in (58). 12 22. It is instrnctive to note that in (21) x, y are simple functions of the integral u = \dxly\ in fact, we find the following results at once : — (1) a>0, i'>0, ax + 5 = i^D . sinh (u »/a), y \Ja = \/D . cosh {u \/a) ; (2) a>0, Z><0, !> = -!>„ a.v + b = a/-D, • cosh (u ^/a), y \la = V-O, • sinh (m \ja) ; (3) a<0, i><0, a = -a„ D = -D^, a^x — b = \/Dj. sin (u^/a^), y '\/a,= \/-D|.cos(M V«,)- From these formulae we deduce, if :;:„ y^, m, are corresponding values of a.-, y, u. >Ja. ^- = iaM\\[\\Ja(u — u^\ ia cases (1), (2), or '\Ja^.'- ^ = tan{^\/a, (m — M,)j in case (3). 23. With the same notation as in (20)— (22), we have the formulae Cxdx 1 , „ b Cdx Cx dx y h rdx ] y ~ a a] y ' 34. By writing x- p^ljv, and then using (21), we can prove that C dx _ 1 , \yq + apx + b {p -\- x') -\- c\ ]{x-p)y~ q ^1 x-p y if ap'' + 25jo + c is positive and equal to q^. But if ajo' + 25/> + c is equal to - q', we reduce the integral to the form r dx _1 _i {apx + b {p + x) + c') ]{x-p)y~ q, ^° 1 q,y j" And if ap^-\-2bp + c = 0, so that x—p is a factor of y', we find C dx \ y ](jio—p)y ap + x — p' 13 25. Apply (21) and (24) to the integrals Cdx r dx where (1) /=a,-+4x + 2, (2) y=-a;'+4A- + 6. 26. (a) If /= (x - a) (.« - /3), where :r > a > /3, prove that Jy = 2 log [V(x - a) + ^{x - )3)!, cf. 21 (a), by using the substitution x = o. cosh" - j3 sinli" ^. When a>j3>x, use the substitution a; = /3 cosh" — a sinh'' ^. (6) Ify'=(a — ;r) (« — j3), where a>^>^, prove that Jf = 2tan-'{y{'^)}, cf. 21 (i) and 22. by means of the substitution ar = asin''0 + /3cos'^. [c] If y^= {x — a) [x - /3), then (24) gives C_dx____ 2 y _ 2 l/x-l3\ ][x-oi)y (3-ax — a~(3 — a.y\x — aj' which should be verified by differentiation. 27. If ^ = tan^.r, then prove that r ^^' —of '^^ Ja + b cos a- J (a + 6 + (a - d) i" ' r ^-"^ — 9 / ^'^ Ja + 6cosx + csina;~' j(a + 6) + 2c^+ (a — 6j t^' Complete the integration in the cases (1) a = 5, 5 = 4, (2) a = 3, f) = 5, (3) a=13, 5 = 4, c = 3, (4) a = 5, 5 = 7, c = l. V. Eatioxal Algebraic Intkgrals. When a rational fraction has been resolved into partial fractions by the ordinary rules of algebra, the integration can be carried out at once, except for fractions of the type 14 where ^ is a quadratic in x with complex factors. Such fractions are reduced to (20) by nsing the formulge of (28) below. If the fraction originally proposed for integration is of the form i2/Z""', where R, Z are polynomials in x, it is usually simpler not to divide the expression into partial fractions, but to assume Z"*' dx \Z"I'^ Z' where Y, Y, are polynomials of degrees 7ik—l, k — 1 respec- tively (k being the degree of Z), with coefficients to be determined. Thus we have the identity E = Z^-nY'^+Y,Z'', ax ax from which the (n + l)k coefficients in Y, Y^ are found. This method is illustrated by (33), (34) below : in each of these we can foresee that even powers of x cannot occur in Y nor odd powers in F,. The same method could be used also for (32), but it is probably quicker to use (28) in this case. 28. Write Z= ax' + 2bx + c, D = ac-b%fixid v=^ dx / X"*\ then prove, by differentiating [Ax + B)IX", and adjusting the coefficients A, B, that we have the following reduction formnlse 1 {ax-\-b . ,, ") ^»=2^i^--+('"-'^«M' Deduce that [x dx 1 \hx-\-c , , \ at 29. Prove that, if ar + c/3 - 2^5' = P, f , , „ ,dx vx-\-q P [ax + b Cdx\ 15 30. By taking w = i in (28) obtain the two standard forms of (35) f da; _ax + l> He dx _ bx -\- c 31. Showthatif«'-a: + l=Z fx dx , /x — 2 \ and find »„. 32. Prove that [ dx 3a;'+5x _ _. / 33. Prove that do- _^ X — x' ..■1_1_ ^2 . , ,a '6 TS" + i tanh- f^) + -^ tan- f^.V 34. Prove that, if Z = x*— 2x''cos2a4 1, / = ril±^- ^ _J_ tan- f^) , J A 2sma \l-a;V' J ^ 2cosa V l+.fV and show that rrfx _ X (.rVos 2a — cos 4a) 1+4 sin^'a 1 f 4 cos'a JX " 4jrsin''2a "^ 16sin'a "^ 16 cos'a ' VI. Irrational Algebraic Integrals. As a general rule we cannot reduce to elementary functions any integral in which the irrational element is more compli- cated than the square-root of a linear or quadratic expression in X. We consider, therefore, integrals of the type where y^= ax'+ 2bx + c. 16 Any algebraic function f{_x, y) is reducible to the form (-p Ti \ J, + -o- ) by the ordinary rules of algebra, where P, Q, li, S are polynomials in x. Thus we need only consider integrals of the form {R dx_ {(rpU\ dx where T is the quotient, U the remainder when R is divided by /S ; we may regard U\ S as expressed in terms of partial fractions, and then the new integrals to be discussed are of the three forms /■ ,^dx C I dx cAx + B dx where -Z is a quadratic with complex roots. It will appear from the reduction formulas (36)-(41) below that these integrals can be reduced to the case n = 0. But the labour involved in the third is almost prohibitive except in the special case considered in (39) ; however, this covers the practical applications which are of principal importance. On account of the fact that we have often to deal with the integrals ly'dx, ^dx/y", it is worth while to give a special formula of reduction (35) for them, although this may be regarded as a special case of (36) or (38). 35. Let y^=ax^+1bx + c, and D — ac — b'\ then verify by differentiation that {n + \)a ly"dx = {ax + 5) y" 4 nD Jy''"W«, ' dx ax + b C dx -^ Cdx ax + b , , , f dx Deduce that if m = [dxjy, then Utandard forms. tdx ax + o Cxdx bx + ci 17 lu the special case b = 0, we note the results (?z + 1 ) J fd.'c = xf +ncj f dx, C dx X , . [dx {d.e jydx = ^{xy + cu), l7=.7' j/dx = }xi/ {2ax'+5c) + ^c'-u, J-^ = ^^ (2ax'+ 3c). 30. If we write u^= \x — , with u^ = u, prove that (w + 1) aM„^, + (2?2 4- ] ) bu^ + '^^z'n-i = '^'''y- Hence show that au^ = y— bu, 2a'^u^ = y (ax — 3b) — (ac — 3b^) u. Thus u^ can be reduced to the form yP^_^{x) + ku, where P,., is a polynomial of degree (?e — l) in ,i', and k is a constant. 37. Similarly, if we write d. ,=\^r and .„=., V ■• jx-y we have (71 + 1) c«„^,+ (271 + 1) bv„+ 7iav,^_^=- y/*"*', and so, in analogy with (36), cv^ = -(yjx + bv), 2c\ = y (3bx - c)la-' +(3b''- ac) v. In the special case c — we can use the reduction-formula to evaluate v, and we find bv = -yjx, (27i + l)bv^ + '2*Vi = "" yl^"*^^ so that », can then be found without using any logarithmic function or inverse tangent. 38. To deal with the integral r dx 18 we write x- p = .-Cj, aud tlieu y'= ax^-^ 2 {ap + h) x^ + {ap^-\- 2bp + c). We write, therefore, a„ b^, c^, x, for a, S, c, :» in the formulEe of (37), where a^=a, b^ = ap + b, c^ = ap^+2bp + c. The reduction formula is {n + I) c^v,_^^ + (2?i + 1) b^v,^ + na^v^_, ^-yfic;"''. The simple case c, = occurs when x—p is a factor of y^, and then v^ can be found without using transcendental functions. 39. li y''=ax^-{-c, X=ra:'-\-s, A = as - re, and the integral to be evaluated is _ r dx the reduction formula is 2(re + 1) (5A)»„^,- (2?2+ l)(a6' + A)»„ + 2a?z2J^_, = -rxy/Z"*'. In this way i\^ is reduced to depend on y,,, which we integrate by the substitution t = yjx ; for then r (/x r dt which is of the type (4) or (5). If the integral is of the type r X dx we write axdx=y dy, aX=ry' + ^, and the integral becomes dy •'/( l(r/+ A)""" which is reduced by (28). 40. Consider now the general case of the integral ^dxl{Xy) where X is a quadratic with complex roots.* We can then * It is at present usual to handle this integral by the substitution v = y/X*; but although the same variable can be used for the two integrals, yet the algebra involved is no less than in the method suggested here, in any practical case. This method has the further theoreutical Objection of introducing a second square-root jX in the result. And according to a general theorem of Abel's, the only root needed is y, whenever the integral can be expressed by elementary functions. 19 always find a, ^ snch that and if we write ^,=y/(x-^), t.^ = yj{x-a), we iiud Hx - g) dx _ ^L_ r dt^ r (x - 0) d.if _ 1 r dt, J Xij - a-0Jrt,'+^' j X^ "ie^ajs^T^^' where A=ps -qr. To find a, j3 when X, y' are given quadratics, we first determine the two values of X for which A'- \y^ is a perfect square in x: these values of X are known to be real and unequal if the factors of X are complex (see Cambridge Mathematical Tracts, No. 3, Oh. I.). Suppose that we have X-xy=A{x-oi)\ X-\f=B(x-l3y, then {\-X^)y'=-A(x-ay-^B{x-i3y, and (X,, - \J Z = - A\ (x - a)" + 5\ (x - j3)'. f dx 41. The general integral ^^^ can be reduced to (39) by writing S, = (x- «)/(.).■ - /3) in the notation of (40). We find then {iix>(3) , J^ , =(oL-(5) ^•'""^^'^^" , where v' = J^f + ?, and (i-g)^_r«-/3y so that we have [ L{x-a.)^M{x-^) _ 1 r (Lg + iy)(l-g r J A'"> '^-(a-^)-'] (rr + ^r''? ^" 42. Apply the formulee of (37) to prove tliat {dx 3a' — 2 f fl?a; iS ^"*j^' '^ y'=*''+^ + i. '.r'y 4. 1- -^ '^ } xy' and evaluate the last integral by changing the variable to 1/a', or by (22). 20 By means of (38) show that and prove similarly that r dx _^y[2x — b) ^r dx and evaluate the last integral by (22). 43. Apply the method of (40) to evaluate the integrals ''[x — \)dx [{x — i)dx nx-\)dx n Xy ' ] Xy ■■ where (1) Z=3«'-10a- + 9, j/'=5;r'- 16^ + 14, (2) jr=3.i;'-10;i-+9, y'= x'-%x -\-lO. 4:4:. Prove that, if a >j3, C dx _ 2 _j //«+^\ ''~J(x + aV(^-+/3)~V{a-^) VVa-jS/' , r^___dx 1 f V(A- + /3) , 1 ''"'^ J(x + a}V(^H-/i^) a-/3 t a- + a ^H J(a'+a)(^+;«)f~~^^ tvi^Tpy^^j ■ 45. Integrals of the types considered in this section can also be reduced to integrals of rational fractions by sub- stitution ; the general method is the following : — Let [p, q) be a point on the conic y^= ax''+ Ihx + c, and consider the other intersection of the conic with a variable line through (/», — q). Thus write t = {x —p)l{y + q), then X, y are rational functions of t, given by U [q + rt) ~ q[\ + af) + 2rt ~ l-at" r-ap-^b, dx . 2dt r. ,„„, and 7 = 13^- «M22). If q=0, the formulae are very simple : and the method has 01 usually been restricted to this special case, but the restriction is not necessary. 46. In the case of a hyperbola (a>0), we can take [p, q) as a point at infinity and use the lines parallel to an asymptote, instead of lines through [p', q) : thus, if we write th en ax + b = ^(v j, t/ ^/a = ^(v + — ] , and — = -; , cf. 21 (a). y sja V ^ ' ExAMPLltS OF SIMPLE PSEUDO-ELLIPTIC CASES. Although an integral which contains the square-root of a cubic (or bi-quadratic) must lead, in general, to elliptic functions, yet it is sometimes possible to complete the integration by means of elementary functions ; such integrals are called pseudo-eLliptic, and a few simple examples are given below. More complicated cases have been worked out by Greenhill with a view to physical applications. 47. 1^ y^ = X [a [x"" + I) + bx] and «J=y/a-, verify that Cx - 1 dx _ r 'idv rx + 1 dx _ r 2dv ]x+l 'J ~ ]v'-b-\-2a' ]x-\ "y ~ ]v'-b-ia' and so evaluate the integrals Cx'—l dx Cx^—\ dx Cx^+l dx Similar methods apply to any integral of the type - — - dx, where /(a:) is a fraction such that /(.?;) +/(!/«) =0. 48. Prove that if v^= x^ + x, then [x-^\ dx .^ , , . rx \/2 = - V2 tanh ' ( — ^ }x-l y \ y rii:i^-=v2tan-'m. }xA-\ y \x^/2) 22 VII. Teigonometrical Integrals. The cases of most common occnrrence are given in (49)- (51) ; these formulEe show that the integrals can always be reduced to (10)-(15) and (27). When the integrand is more complicated, but is expressed as a rational fraction in sin a,- and cos .7;, the integral can always be transformed to an algebraic fraction by means of the substitution , , . 2t 1 - f , 2dt t = tan -^x, sin x = ^ ^ ^^ , cos x = ^ ^ , ax = -, l+f' 1+t" 1 + f But it is often possible to bring the integral to simpler forms by taking as the new variable tauA-, or sin a;, or cos*-, according to the form of the integrand ; for instance, when the integrand is a rational fraction in tanj;, we should use tana; as a new variable rather than tan^a:. In other cases we can use trigonometrical transformations to bring the integrand into a form similar to that of partial fractions (for algebraic fractions). A common type is the fraction P^(sina:, cos a;) nsin(x — a) ' where P^ is homogeneous of degree r in sina; and cosa;, and there are n different factors in the denominator. Here there are two types of partial fractions, when ji > ?-, (1) ^—. — 7 r, when w - r is odd, sm(a; - a) (2) S-; ; r, wheu w — r is cvcu. tan (a;— a) The value of the coefficient A is found by multiplying by sin(a; — a) and then writing x = a; this gives A= P^(sina, cosa)/n'sin(a — j3), the accented n' containing (?z — 1) factors, in each of which a comes first. 23 The proof that such resolutions are possible, and the necessary additional terms when n^i\ are ionnd most quickly by using the complex variable ^ — Qix — COS g; ^ J sin g;_^ and then applying the ordinary rules of algebra. For instance, if r = n, we find the equation P„(sinas cos«)_^^^ A n sin (a' — a) tan (,« — a) ' where the value of A is given by the same formula as before if C is the real part of e"'P^(l, t) and o-=Sa. In particular, if P^(sinx, cosx) = n sin(x— ^), where there are n factors in the product (not necessarily all different), we find from this formula C = cos (S6I - 2a). When the values of a are complex, as in (53) below, this method is not satisfactory, and the use of a new variable is simpler. 49. If s = sin.r, c = cos.t', verify that (m + n) [s'^d'dx = s"'"'c"-'+ (w - 1) ^s'V-'clx = - .s'^''c"*' + im - 1) js"'-'c"dx, (»« - 1) j^ «?^ = - ^1 - (?« - 1) Jp;Pi ' 50. Useful special cases of (49) are given by taking »2 = 0, M = 0, or m = n; then, with i! = tana-, we find the following six cases : — s in" X 71 Is" dx = - s"-'c +(n-l) ^s'"' dx, cos" X n Jc" d.L- = + sc"-' + (w - l ) Jc""' dx, dx 24 ■' n-l ' sec x (?2-i)J— = _ + (re-2)j^-^, cosec ^ (re_l) __ = __j + (^_2) U^. ] & S J s 51. If v^ = '\dxjX"*^, where X=a + 5cosa;, verify that [n+ 1) {a'-F) w„^,- (2?2+l)a»„+?z»„_,= -5sina,-/Z". More generally, if ^ = a + 5 cos « + c sin :r, we have (n + 1) (a=- b'- c') »„„- (2w + 1) a»„ + wVi = (— 5 sin ,r + c cos a;)IX''. Alternative methods of transformation are suggested for these (and other similar) cases in (55). 52. Reduce to partial fractions and so integrate the fractions P/sin(:c — a) sin(x — j8) and Q /tan (a; — a) tan (a; -j3) where (1) P=l, or sin«, or sin^x, (2) Q=l, or tanx, or cot^r. If P is sin^^•, prove that the first fraction is equal to . , „ , 1 f sin'a sin'jS ] sm(a — /y) [sm(A- — a) sm(a; — /S)j Integrate also the second fraction by using the variable t = ta,Txx, and consider further the case with Q = tan'x. 53. By taking i; = tan-|x and applying (34), or » = cosa; and using (40), prove that (2cosa; — l)(^a; 1 > / 2i! a/2 ' r (2cosx — l)dx _ 1 /2ts/2\ J 2-4cos«+3cos'a' " V2 ^^ \1 + 3f) i-tanh-( ^^^^°n , V2 \2-cos«;' 25 r (2 - cos a:) dx _, / 2^ \_ _j / sin.r \ J2 — 4cosa- + 3cos"a;~ ^^ \1 — 3tV " ^^ V2cos:j. — ij ' Similarly use the variable » = sinh.r to evaluate f (1 +2sinh;i-)f/x , r (2 — sinh a;) ^a^ J 5 — 4 siuha; + 2 sinlfa; J 5 - 4 sinha? + 2 siuh'j: ' 54, Fiud formulEe of reduction for ^cosnxcos'"aid.v, ^cosna:sin"'x-dx, etc., and verify that they can be brought to the forms (m + n) Je'"^ cos"' a: dx= — le"'" cos™.r + m Je'("-i)^ cos"*-! x dx; {m + n) Je'"* sin™ xdx = — te'"^ sin™ x-\- im Je'(''-i)« sin'"-! x dx. So also prove that f e'«^ , fe'(«-l)« fe'(n-2)a; ax = 2 ;— dx — dx, Jcos'^x Jcos™-!^ Jcos™.t' f e'"" f e'(''-i)»^ fe'C»-2)a: ^ a« = 2t ^ ; — a*' + —. dx, J sm'" X J sm™-! a; J sm™ x 55. If (a + b cos 6)(a — b cos (f) = d'— b'\ where a > and a'>b'', and ^, ^ both lie between and v, verify that dd d(h . sin sin ^ and - « + 6cos^ '^(a^—b''), a + b cos 6 'J{a^—b'')' If (a + b cosh u) (a — b cosh v) = d'—b'', where a>0 and a' > 6\ and u, v are both positive, verify that du dv , sinliM sinh» and a + bcoshu >J{d'—l)') a-\-bcoshu '^{d'—b')' If (a + bsmhu)(a — bsmhv) = a'+b', verify that, when a + b sinb u is positive, du dv , coshw cosh» and a + 6sinhM s/{d'-^ b'') a + bsi\ihu -^{a^+h')' Consider the various cases which arise when V > a\ and either (a + ^cos^) {a — bcQshv) =a'—b', or (a + bco&hu){a — bcos^) = a^—b''. 26 VIII. Definite Integrals. The followiug integrals can all be evaluated by the fundamental foi-mula j f{x)dx = F{b)-F{a), if \f{x)dx = F{x); but in (58) the preliminary transformations of p. 12 may be useful. In the numerical examples (60) it should be remembered that logarithms are taken to base e, and that angles are expressed in circular measure. It is sometimes easier not to use the tables, but to calculate by means of series such as tan"'^- =x-\x^+\x^—..., i'A.n\r^x = x+^gx' + j^x^+... 1 x' 1.3 x^ 1.3.5 x' sm-. = . + --+— -+^-^g- + ... . ,_, 1 x^ 1.3 x' 1.3.5 x' smh x = X — I- — — I- . . . 2 3 2.4 5 2.4.6 7 which are obtained (whenx' 0. But if ac-V——p^, we get a similar formula with tanh"' in place of tan"' and ^, in place of p ; only care must be taken to see that both roots of the quadratic fall outside the range of integration. 58. If y'=az"+ 1hx-vc, prove from (22) or (45) that r^ = Atanh- p--"°-^^i , if a>0; when a is negative, we replace sja by s/^—a) and tanh"' by tan"'. Also r'_^ = ltanh-I ^1^^ \ if ap'+2bp-\-c is positive and equal to q' ; when this expression is negative, we use V(- {ap'+ '^bp + c)\ and tan"' in place of tanh"'. In applying this formula it is important to note that p must not fall between x^ and .f,. 28 59. If a, b both lie betweea and 1, prove from (58) that J _, Vi(l - 2aA' + a') (1 - 2bx + b'\ 'J[ab) and evaluate the integral if a > 1 > ^ > 0. 60. As exercises on numerical work prove that [ V(-i-' + 4)(^:r = 8-96, [ V(^'- 4) rfx = 6-89, i ^/(4:-x'')dx=l■n. ■' 61. Prove that riir rijr sin^« dx = ^TT = cos':i- dx. Jo J a This result is of constant application in physical problems : it is conveniently stated in the form : — The mean value of sin'iu or cos'"'*- (integrated over any multiple of ^tt) is \. 62. If m, n are unequal integers, prove that smmxamnxdx = (i, j cosmx cosnxdx = 0. ■' ■'o r*" • 2„ 7 1.3.5...(2w-l) TT 63. sin'"xdx = ^ ' J '^ ■' 2.4.6...2re 2 r*' • »„.. J 2.4.6...2W sm'" 'a- dx=--—- — J- -T , J„ 3.5.7...(2n+l) ' where n is an integer ; these follow at once from (50). 64. Prove that the integral riir f{m, 71)= sin'"a;cos''a;£^a; J satisfies the relationy(»2, 7i) —J(n, m) ; and that (49) m + n)/'(m, n) = {m— '^)f{m, — 2,n) = (n — l)f{'m^ n - 2). 29 Deduce that when m, n are both even integers but that when m is odd f(m, n) = i . ■^ ^ ' ^ {n + l){n + 3)...im + n) "When n is odd, there is a similar formula which can be found by interchanging m, n. 65. Prove from (54) that C08"'a: COS w A- dx = cos"' 'x cos (w — i ) jj: c^x, Jo m + n}^ and so find the integral when vi'>n. In particular cos"a.- cos nx dx = ■ /: 2"*' adx 67 I -^ — ^ = ±i's-, according to the sign of a. I ^ — ^7 = - tan"' \\\ , if »' = ac - V, J„ ax'+25.r + c /» \hl ^ where a, ac — V are positive and the angle lies between and IT. Hence also r dx -n J -00 «*•'+ '^hx + c p ' 68. If Z= a;r' + 25a; + c, and «„ = f dxjX"'', it follows from (28) that (2?2-l)a r"a;«?.r_ (2??- 1)6 l)a C xdx _ I 2?2/ "-" J_„JL"'' Inp' ""' Thus we find from (29) and (67) J -00 JL ip 1.3...(2w-l) Tra" ^""^ ^"^ 2.4... 2« 7^'- 30 69. (a) If X=X+x*-2x^ cos 2oL, where 00, we have f" dx — I C ^^^ 1 ^^ J „ [ax' + 2bx + c)tf ~ ^*J „ (a:r' + 26a7 + c)l ~ s/(ac) + i " (5) Prove from (58) that, if 00), „_ r^" cos''xdx c_[*' sin'xofx J „ p^ cos'' a- + g'' sin'jc ' ] o P' cos^r + q' sin'' a: prove that C + S = ^Trl(pq), p'C +q''S=^iT^ and deduce that pC=qS=^Trl{p + q). Show that, if c>a>0, ['' ^/(a'-x')dx IT 6. If a, (5 are positive and r' = a' + b' — 2ab cos:!:, prove a —b cos.r) that r d {b) sin^^ - •'0 , , r . dx W J^ sin^ — [d] {a — b cos ^) sir where in each case the first value is to be taken when a>b, the second when a < §. r" a r b 77. e''^cosbxdx = ~^^rj-j^, J e''^ sinbx dx = ■^^-^, , where a is positive. These two are contained in the single equation [ e'^^dx^- (if c = a + tb). Jo ^ ax IT or 0, 2a' °^ 26' ' 2 a 2 or ^, dx 2 r a or 0, ■' 32 71 ^ 'dx=-:^^ if a>0. a This result is valid when a is replaced by c=a + t5; but a simple proof requires more advanced methods than for (77). _„ r°° dx TT .„ 79. — =-— =— if a>0. J„ coshaa- 2a IX. Integration of Series. It is impossible to give any detailed explanation here of the conditions under which term-by-term integration of an infinite series is permissible : some of the more useful tests will be found in my book on Infinite Series (Arts. 44, 45, 175, 176). But the majority of ordinary cases can be tested by the simple rule : — If for all values of x from a to b, J„{x) is numerically less than i/„, where M^^ is a positive constant, and if the series Slf is convergent, then the equation fjV.{^)]da: = 2j''/Jx)dx, will be correct. Thus for instance the two series 1 — f (a) — --, = l + 2t cos X + 2f cos 2x + 2;;' cos 3.!; + . . . , t SIQ X (B) - — 5= tsmx+ <'sin2x+ f sm3x + ..., ^' ' 1 -2tcosx + t in which 0, q being positive) ^'^ = 1 - 2icos2j:-f 2i;*cos4A'-... , p cos x-^ q sin x where t=[p — q)/[p-{-q). Hence obtain (74) and (75). 82. Deduce from (/3) that if n is an integer r • / tsmx \ „ sin nx — -, ax = -kirt . J„ \l-2(tcosa; + i;7 Hence prove that if a', 6' are both less than 1, r sixi'wdx _ ""■ J„ (1 +a'-2acos«)(l+6'-2(5cos:i-) "" 2{l-ab)' 83. Deduce by differentiating (a) that (if <^0, a' >5*J by writing a = c{l +f), b = - 2ct, so that ^/{a'-F) = c{\-f), aud 2e=a^ ^J{a.' -h'). Thns we find for instance r log {a + b cos x) dx = IT logc, — irb , f sinxcosa; , — tto J „ (a + i cos x)' icW{a' - 6') X. DlFFBKENTIATION AND INTEGRATION OF 1nTE!GRALS. From any integral containing a parameter we can obtain others by differentiation or integration with respect to the parameter ; but difficulties sometimes arise in the case of infitiite integrals (that is, integrals which extend over an infinite range, or which have an infinite integrand). For some of the more useful working rules in such cases, see my Infinite Series (Arts. 171, 172, 177] ; but even without a knowledge of these tests we can still regard these methods as useful for suggesting results. For example, (78) can be derived from (77) by differentiation with respect to a ; but this could not at present be regarded as a complete proof, because the range extends to infinity. In like manner the results of (68) are suggested from (67) by differentiation with respect to a, b, c. It has sometimes been suggested that the method of differentiation should be applied instead of formula? of reduction to cases such as (28), (29), (38), (41) : but in general this process is tedious and in the case of (41) it is hardly practicable. 86. By differentiating (73) with respect to a and b, prove that r dx Tra J „ [a+bcosxf ^ [a:' - b')i ' r cosxdx %'nab {a + bcoBxY {a' -by sin' a: cos .r , —Trb — dx = /„ {a + b cosa:/' ' ^c'^J[d' — b'') ' where 2c = a + \J [a" — b') ; compare (85). 35 87. Verify the accuracy of the equation (85) I log (a + 5 COS .«) = TT log c, by differeatiation with respect to a and b. Deduce also (76) [d) from (76) (c). if i^(x) is a polynomial of degree not greater than n, and c>6>a. XL Simpson's Formula. 89. If y is a cubic function of «, and y,, y^, yj are the values of y at the beginniug, middle and end of an interval of length I, then the integral of y over this interval is equal to y(y> + 4y, + y3)- 90. If y is any function f[x), which does not vary rapidly in the interval I, the last formula can be applied as an approximation to the integral ; the error involved can be put in the form 2880'^ ^^'' where 5 is some point within the interval. Thus if f" {x) is positive throughout the interval, the approximation will be too large. 91. Apply (90) to calculate the values of the integrals in (60), and estimate the order and sign of the error in each. 92. The approximation can be improved by dividing the range of integration into n parts and applying (90) to each of them. In this way the interval I is divided into 2m parts, and the values of y must be found at each point of division ; we denote these values by y,, y,, y,, ..., y,„^„ and then the ap^proximation to the integral is 36 The error is less than Ml^l(28S0n*), if M is the greatest numerical value oi f"{x) in the interval. 93. An important application of (90) is to find the area of a segment of a circle of large radius. If h is the height of the segment, and I the length of the chord, we can write y, = 0, y, = h, y, = : and the rule then gives as the approximate area A = llh. The exact formula is J = \r'Q — ^{r — h) l\, where '2rh = lV-^h' and tanJ0 = 2A/^. We can estimate the accuracy of the approximate formula by expanding A in powers of t=2hjl, which gives ^ = ^ (^ + rXI - 37577 + 57779 —•)• Thus if k does not exceed ^l, the approximation will generally suffice, for practical calculations. For such segments the original form of A is not very convenient in numerical work ; because A is given by the difference of two nearly equal areas. 94. Another practically important application of (90) is to the calculation of volumes and centroide. If a solid is obtained by the revolution of a straight line, or of an arc of a conic (about one of its axes), the area of a section at distance x along the axis of revolution is expressible in the form S^ax' ■\-2bx+ c, and so the exact volume is given by V = ISdx = \l (S, + S,+ AS,), where S^, S, are the areas of the ends, and S, of the middle section. When the value of the coefficient a is known, we may use the equivalent formula V=ll(S, + S,)-ial\ derived from 3^ = I (S^ + S,) — \al\ The centroid is the same as that of 3 particles propor- tional to -Sj : 4*Sj : S^ placed at the corresponding points on the axis of revolution. For a cone (of angle a), a=7r tan'a ; for a sphej'e, a=—ir\ for a paraboloid^ a = 0. 95. The formnlaa of (94) apply also to a solid bounded by planes, two of which (the ends) are parallel ; it is then necessary to interpret I as the distance between the ends. Examples are given by a mound, with horizontal top and bottom, and plane faces ; or a straight embankment on a hill-side, in which the ends are vertical planes perpendicular to the length of the bank. Another case is a tetrahedron in which two opposite edges are regarded as the ends. XII. Planimeteic Formulae. 96. When a rod of length I receives a differential displacement, prove the equation dS,-dS=l{.d