straight line of a given length extending from the axis of x to
the axis of y.
Let c be the length of the line, and a and h be the inter-
cepts on the axes of x and y respectively ; then the equation
of the line is
and the equation connecting a and h is
a« + b^ =6-2 (2).
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312
DIFFERENTIAL CALCULUS.
Differentiating (1) and (2) with respect to a and 6, a being
taken as independent, we have
X ^ y db ^ .db
and therefore, according to Art. 128,
X y X y
^ _ ^ _ a"_ ^ ___ 1 .
V~ T "^ ^ "" F " c^ '
whence a =. xid, b = y^ct, and
is the equation of the envelope. The
figure represents the curve traced in
the several angles of the co-ordinate
axes.
Ex. 2. Find the envelope of the se-
ries of ellipses formed by varying a
and b in the equation
a and 6 being subject to the condition ab = c'. By differen-
tiating with respect to a and 6, regarding a as independent,
we have
x^ y^db „ 1 . 1 d& „
a» "^ 6' da ~ ' a "^ 6 da '
2 2 1
whence
c*
which is the equation of an hyperbola referred to its centre,
and asymptotes as axes.
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EXAMPLES. 313
EXAMPLES.
1. What is the radius of curvature of the curve
y = a;* — 4a?'— 18a;?
at the origin of co-ordinates ? . 1
Ans. p = —-•
2. Find the parabola which has the most intimate contact
with the curve y = -^ at the point having a for its abscissa,
the axis of the parabola being parallel to the axis of y.
3. Show tliat, at one of the points where y = in the curve
2 _ ox (a? — 3a)
y "" x - 4a '
the radius of curvature is - ; and at the other, -^,
o Z
4. What is the radius of curvature of the spiral of Archime-
des, the polar equation of this spiral being r=zad?
(aH-^
Ans. p = dz -^^ —.
5. The Lemniscat^^ of Bernoulli is the locus of the points in
which ' the tangents at the diflFerent points of an equilateral
hyperbola are intersected by the perpendiculars let fall upon
them from the centre of the hyperbola. Its polar equation is
r' = a^ COS. 26. What are the radius of curvature and the
chord of curvature at any point of this curve ?
Ans. p = X- ; chord of curvature = ^ r.
3r' ^ 3
6. If a curve have y =z-(e^ -\-e c J for its equation, prove
that the general co-ordinates of its centre of curvature are*
40
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314 DIFFERENTIAL CALCULUS.
7. What is the envelope of all ellipses having a constant
area, the axes being coincident ?
Ans. 4x-y2 =: c^; nc^ being the given area.
8. Find the envelope of the curves represented by the equa-
tion
M+(^?-7-.
a and h being the variable parameters connected by the equa-
tion
mh
y^
^"^•P + P = ^-
9. Find the envelope of the system of straight lines con-
necting, pair by pair, the feet of the perpendiculars let fall
from the diflferent points of an ellipse upon its axes ; the equa-
tion of the ellipse being
10. What is the envelope of the series of circles, the circum-
ferences of which pass through the origin, and which have
tlieir centres on the curve of which the equation is
Ans. (^2 + y2 _ 2axy — ia'^x^ — 46^^ = 0.
\ .
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INTEGRAL CALCULUS.
-<=sX=^
SECTION I.
MEANING OP INTEGRATION. — NOTATION. — DEFINITE AND INDEFI-
NITE INTEGRALS. — DIRECT INTEGRATION OF EXPLICIT FUNC-
TIONS OF A SINGLE VARIABLE. — INTEGRATION OF A SUM. —
INTEGRATION BY PARTS. — BY SUBSTITUTION.
191. Any given function of a single variable may always
be regarded as the differential co-efficient of some other func-
tion of the same variable ; that is, there is some second func-
tion, which, when differentiated, will have the given function
for its differential co-efficient.
For let/(a:) be the given function. If this admits of possi-
ble values for real values of rr,
we may construct the curve
CPD, which, referred to the
rectangular axes Ox, Oy, has
7j =/{x) for its equation. The
area included between this
curve and the axis of Xy that is
limited on the one side by the fixed ordinate CA, correspond-
ing to x = a, and on the other by the ordinate PM, corre-
sponding to the variable abscissa a?, is evidently a function of
815
I
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316 INTEGRAL CALCULUS.
x; and, of this function, y orf{x) is the diflferential co-eflScient
(Art. 164) : hence we should have
^(area^CPif)=/(x),
, (area A CPM) dx =/{x) dx.
192m It will be found that the operations of the Integral
Calculus are mainly those of passing from given functions to
others, which, by differentiation, would produce the given
functions. The fact that these operations are the inverse of
those of the Differential Calculus has been taken as the basis
of the definition of the Integral Calculus. But the fundament
tal proposition of the Integral Calculus is the summation of a
certain infinite series of infinitely small terms. To effect this
summation, we must generally know the function of which a
given function is the differential co-efficient. The proposition
may bo stated thus : —
Let /{x) be a function of cc, which is finite and continuous
for all values of x betvreen Xq, x^, and of invariable sign be^
tween these limits. Let Xj^ be greater than Xq, and divide the
difference x^—x^ into a number n of parts, equal or unequal,
represented by a?i — iCo, x^ — x^j x^ — x^-.^j a;^ — a?„_i; re-
quired the sum of the series
S=f(Xo) (xi-Xo) +/(^i)(^i - ^i) H
h/(^n-i)(a:„-a;„_i),
'when the number of parts into which x^ — Xq is divided is in
creased without limit, or n is made infinite. For brevity,
denote the intervals Xi—Xq^ X2 — Xi.,,, a;„ — a?„_i, by
^1, A2 . . ., A„, and the series becomes
S=Ax,)h,+/{x,)h,+ ... +/(x„_,)A„_,+/(x„_,)A, (1).
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MEANING OF INTEGRATION 317
Now suppose F(x) to be the function of a?, of which /(a?) is
the first derived function ; then
But, before passing to the limit, we should have
F{x + h)-F{x) ,. , , ,. . ,^,
""^ — h =/(^) +P (A.rt. 15),
p being a quantity that vanishes with A ; therefore
Fix + 7t) - nx) = h \/{x) +p\ (2).
In (2), giving to h the values hu h^.^.y A„, and to x the
values a^oj a?i..., a;„_i, a;„, and denoting the corresponding
values of p by Pi , P2 • • •> Pm observing that
XQ-\-hi'=- Xiy CCj -|- ^2 ^^ ^2 • • •;
we have
F{x,)-F{x,)=h,\f{x,)^P,\,
F{x,)-F{x,)=h,\f{x,)+p,\,
F{x„_i) - F{x„_,) = A„_, \/ix„_,) + p„_i I ,
FiX„) - F{X„_,) = K \f{Xn-{) + P» j •
Adding these equations member to member, for the first
member of the result, we have F{x„) — F{xq). The second
member is composed of two series, the terms of one being of
tlie form h/{x) ; and of the other, hp. Denote the sum of the
terms of the first series by 2/(^x)h, and of the second by 2ph;
then our result may be written
F{x,) - F{xo) = Zf{x) h + Iph (3).
If p\ the greatest among the quantities Pn P2 • • • ? P« , be sub-
stituted for p in the series represented by Iphj we should have
^ph <P^(Ai + A2 + • • • + AJ = p^{x, - X,).
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318 INTEGRAL CALCULUS.
But p' vanishes when Ti is decreased without limit: hence
Fix„) - Fix,)
is the value towards which the scries Jf{x)h converges when
the quantities of which h is the type are diminished without
limit ; that is,
\im.2/{x)h = F{x„)^F{x,) (4).
193 • It may be readily proved that 2f[x)h has a definite
value when h is indefinitely decreased, and when, therefore,
the number of parts into which the interval x^—x^ is divided
becomes infinite. For let A^ be the least, and Ai the greatest,
of the values assumed hj f{x) for values of a; between cCq, x^:
then
2f{x)h >A, {h, + /i2 H h A„) = A, (x^ — Xo\
-y(x)/i<^i(/ii + /i2H [-h„) —Ai{x^-Xo)',
and since, by hypothesis, both Aq and A^ are finite, the same
is true o{ 2/{x)h. It is evident that the values o{/{x) inter-
mediate to Aq, Ai, will be furnished by the expression
being a proper fraction ; and that such a value can be as-
signed to as will make
2f{x)h = {x^ - Xq)/\ Xo + (x„ — Xo) j
a true equation.
194. Putting Eq. 3 of Art. 192 under the form
2/(x)h = F{x„) - F{Xo) - 2ph,
it is seen that the value of 2f(x)h will, in general, depend on
the number and value of the parts Ai, A2 • • ., A»> into which
the interval x^ — Xq is divided, but that lim. 2/[x)h, for which
2ph vanishes, is independent of the mode of division. When
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DEFINITE AND INDEFINITE INTEGRALS, 319
all the parts into which x^ — a;© is divided are equal, each is
equal to -^ ; and any one of the intermediate values of Xy
as x^, is equal to a?o + - {x^ — x^). In this case, the value of
\iT0i.2f{x)h is represented by j V(^) ^^ = ^(^n) — ^(^o)-
The symbol J signifies sum, and dx represents the A = Aa? of
the expression If(x)h. The quantity
fy{x)dx = F{x„)-F{x,)
is called a definite integral ; the operation by which we pass
from/(x)c?x to i /(x)dxh called integration; and a;„, Xc,
are the limits of the integral. Since F[x„) — F{Xq) is the
value of this definite integral, we must first find the function
F{x) of Xj of which /{x) is the difierential co-efficient. The
relation between /(a;) and F{x) is expressed by
fix) = ^- Fix),
which, by the notation of the Integral Calculus, is
f/ix)dx = Fix).
195. The function F(x) of x, which, dififerentiated, would
reproduce f{x) dx, is denominated indefinite integral. But
a constant connected with a function by the sign plus or mi-
nus disappears in difierentiation ; therefore the more general
relation be£ween/(x) and F(x) is
Jf{x)dx = F{x)±C:
so that the proper value of y to verify the equation
is given by the equation
y = Jf{x)dX:^C;
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320 INTEGRAL CALCULUS.
and the two symbols d and /, the one indicating differentia-
tion, and the other integration, neutralize each other, and we
shall always have
Jdu =iU::t: Gy djdu =z du.
The constant thus added to an indefinite integral is called
the arhiirary constant of integration^ or, simply, the arbitrary
constant; it being any quantity which does not depend on the
independent variable x.
The operation of passing from an indefinite integral to a
definite integral consists in substituting in the indefinite, suc-
cessively, the limiting values of the independent variable, and
taking the difference of the results. The arbitrary constant
will, of course, disappear in the subtraction.
19G* In differentiation, constant factors may be written
before the sign of differentiation. The same may be done in
integration. For
fdau =■ aUj a Jdu = au ;
fdau = a Jdu, or fadu = a Jdu ;
or, more generally,
fqf(x) dx = aj/{x) dx.
Observing that ]f{x)dx is the expression for the limit of
the sum 2f(x)LXj that is, the expression for this sum, when
the number of parts of the interval x^ — x^ is increased with-
out limit, and the value of the parts severally correspondingly
decreased, it is evident, that, at the limit, the addition'or omis-
sion of a finite number of the components /(a;) a a; of 2/(x)ax
would not affect the result.
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DIRECT INTEGRATION, 821
A single term/(a;)Aa; of the expression -2/'(x) a rr is called an
dement.
197* Direct integration of simple functions.
We shall, for the present, confine ourselves to the deter-
mination of indefinite integrals, to which it must be understood
that an arbitrary constant is to be added.
There are many cases in which a function is at once recog-
nized to be the differential co-efficient of another. In such
cases, we have simply to write the second as the integral of
the first.
Subjoined is a table of the integrals of the simple functions.
J n-\-V J W
fsin. xdx = — COS. Xj Je'^dx = e"^,
^dx
f COS. xdx = sin. x, j — = Ix,
/dx ^ r dx . , ^ 1 a?
Y- = tan. X, I / » ' "I = sm. -'-== — cos.""* -,
cos.^ X ' •/ \^a^ — x^ a a!
/dx r dx , ^
r—7- = — cot. X. I ., ., = sm.-^ a; = — cos.^' x,
sm.^x ' •/ V 1 — x^
r^^^^ltan.-^==^lcot.-^^.
Ja^ + x^ a a a a
In all of these formulas, x may be the independent variable,
or it may be any function of the independent variable ; for if,
,11+1
r X '
in the formula Ja;"rfa; = — xT; ^ l>© replaced by f{x), we
should have
/j/(x)rrf/(x) = L^|i^.
//j^n-f 1
x'^dx = - — r— :i- reduces to
n+1
= oo;
41
rdx _ 1
J'x'~ 6
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322 INTEGRAL CALCULUS.
/doc
— = Ix, The failure of the formula
•4/
to give the true result in this case arises from the fact that
the transcendental quantity Ix cannot bo represented by an
algebraic expression. It may, however, by a suitable trans-
formation, be made to give the true value of Jx^^dx when
n = — 1. Take the general formula Jx^'dx = ^ + C,
which may be written
Now, the term A t-^ , in the second member, may be in-
' ' 71 +1
eluded in the arbitrary constant C: and thus we have
x"dx= -y-^ r-^ + = 7-1—+ C;
•^ n + 1 n+l 71 + 1 ^ '
or, omitting the constant,
r n^ x"-^' - l ,
x"ax =i T-^ — = - when n = — 1.
•' 71+1
The true value ot this is found by differentiating the nume-
rator and denominator with respect to 7i, and taking the ratio
of the differential co-efficients (Art. 101). We find
/x«+^ — 1\ /x^'-^HxX
n+l /„=-i\ 1 /„=_i
IDS. The rules of the Differential Calculus enable us to
find the differential co-efficients of all known functions ; but
the inverse operation, of deducing the function of which a
given function is the differential co-efficient, is not always pos-
sible. Whatever the assumed function may be, there must be
some other function of the quantities involved, which, differ-
entiated, would produce it (Art. 191). The second of the two
functions thus related as differential to integral may not be-
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INTEGRATION OF A SUM, 823
long to any of the small class of simple functions which have
been admitted into analysis, or to any combination of such
functions ; in which case, we are limited to series and approxi-
mations for the expression of integrals. For example, we rec-
1 X
oernize . - -, to be the differential co-efficient of sin.^^-,
/dx . X
—7— = sin.""^ -, because the latter function has
been named, and its properties investigated. Had this not
/dx
— r— == could not have been ex-
Va —^
pressed by means of a simple function.
199. Integration of a sum of functions of the same vari-
able.
In the Differential Calculus (Art. 19), it is proved that if
then ^1 =fi^x) i cp^x) ± V^^ (x) ± . . . ,
or dij =^f'{x)dx^ ^' {pz)dx ^'\\)' {x)dx , . .:
whence
Jdy = y =^ ^f'{x)dx =b f(p^{x)dx i fip^{x)dx ....
Hence the integral of the sum of any number of functions is
the sum of the integrals of the component functions. For ex-
ample,
also
/(5:r*— 7x3+ 4a; _ 3) j^ _ ^5 _ ^ ^4 ^ 0^2 _ 3^^
and / ( • — ] dx=z-x^^ 2x^ + 5?x.
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324 INTEGRAL CALCULUS.
200. Integration by parts.
If u and V are functions of the same variable, wo have, by
differentiation,
d{uv) __ ^^ !_ ^^
dx dx dx
The integration of both members of this gives
dv ^ , r du
therefore
uv =fu -J- dx +fv -J- dx:
r dir . /• rf^ ,
Itc-r dx = uv — Iv-r- dx.
dx "^ dx
or Judv =■ uv — fvdu.
This method of integration, by which the determination of
an integral Judv is reduced to that of another fvdu, is fre-
quently employed, and is called integration by parts.
Ex. 1. fx^cos.xdx.
Put x'^ =z Uj COS. xdx = dv=: d sin. a;; then, by the formula,
fx^ COS. xdx = fx^d sin. x =: x^ sin; x — 2fx sin. xdxy
Jx sin. xdx =z — fxd cos. x = — x cos. x -\- J cos. xdx
= — X cos. X -\- sin. X.
We shall therefore have, by the substitution of this value in
ihe first integral,
fx^ COS. xdx =: x^ sin. x -{-2x cos. x — 2 sin. x,
Ex. 2. flxdx.
Make Ix = u, dx=dv; then flocdx = xLx — x.
Ex. 3. Jx^'e'dx.
Making a;** = w, e'dic = cZv, the formula givi>#
^x^t^dx = ic'^e' — nfx^'^^e'dxj
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INTEGRATION BY SUBSTITUTION. 325
and the integration of x*^e^dx is thus brought to that of
x'^-^e^dx. By another application of the formula, the expres-
sion to be integrated would become x** ~^e^dx; so that, if n be
a positive whole number, the proposition would be reduced,
after n applications of the formula, to finding the integral
e^dx = de^. 'Hence, by a series of substitutions, we should
have the required integral.
Making n = 1, Jxe'dx = e'{x — 1),
« n = 2, Jx'^e'^dx = oj^e* — 2fxe''dx
z=6*(a?^~ 2a; + 2).
201. Integration by substitution.
It is sometimes the case that a diflferential expression,
f{x)dx, which is not immediately integrable, becomes so by
replacing the independent variable by some function of a new
variable. The function selected must be such that it shall be
capable of assuming all the values of the variable for which it
is substituted within the assigned limits of the integral.
Let t be the new variable, and suppose x-=(p{t) ; then, by
the Differential Calculus,
dx
— = (f}\t)y or dx = cp^{t)dtj
dt
and f{x)dx =f{(p(t)}(p'(t)dt:
whence, by integration, ^
Jf{x)dx = ff{cf(t)W{t)dt;
in which it must be remembered, that, if the first integral is to
be taken between the limits a and i, the second is to be taken
between the corresponding limits a^ and h\
Ex. 1. f{ax + hYdx.
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326 INTEGRAL CALCULUS.
Put ax-\-h=ztj whence dx = -dt ; and therefore
a
1 1 /» + !
f(ax + bYdx = - ft^dt = - - — -.
•^ ^ ^ a*' a n+l
Replacing t by its value, we have, for the required integral,
[(ax + hydx = - ^ ! — ^
Make 8x' + 5 = ^, then Zx'^dx = - rf^,
o
and
therefore Wrl^ = l^i'^' + ^)-
It is evident from these two examples that success in eflFect-
ing integration by substitution must depend on the ingenuity
of the student, and his knowledge of the forms of the diflfer-
entials of the simple functions.
Miscellaneous Examples.
/xdx
-,-, ^. -Ma\^QVa' + x' = l: .'.a^+TC^ = t\
Va^ + x^
xdx = tdt ; and therefore
/xdx -
Va^Tx'^=f''' = ' = ^-' + -''
2. Js/a'^x'^dx.
Putting \/d^ — x"^ =^u, X =1 Vj and integrating by parts
(Art. 200), we have
fV^^^ir^^dx=x^^[^zr^2^r ^y^ .^.
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MISCELLANEOUS EXAMPLES. 327
-dx
x^
But Js^a" - x' dx = r ^ f
Therefore, by the addition of (1) and (2),
2/>v/a2 — a?2 ^^ _ icVa' - a;^ + a^ f ^^ .
and since
a^/ f ^ =o»8iD.-'g, Art. 197,
we have finally
•^ 2 '2 a.
-7====. Make\/a;*-^ + a=^=^— ic; .-. a2 = ^2_2^a;.
f X
Hence, by differentiation, dx = — - — dt : therefore
r dx _ r 1 t — x - __ rdt __ ^
= l{x + \/x2 + a2).
-y===g. Making \x'^ ^ a^ z=z t ^ x, and proceed-
ing as in Ex. 3, we should find
5. js/x'^ + o? dx. Integrating by parts, Art. 200, we have
[* x^-dx
fs/x^ + a* dx = xs/x' + a2 - f ^^j^ (i).
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328 INTEGRAL CALCULUS.
But /V^^H=^^^=r4=^^^
-r ^^^^ , ..c dx
Therefore, by the addition of (1) and (2),
By Ex, 3,
•^ V a? + »
and hence
■I ^2
/V^' + «' dx = 2 a? Va;' + a^ + ^ Z(a; + Va;'^ + a').
6. /\/^^"=^ ^^ = I a;Vaj2_«2 _ ^ ;(^ + Va;^ - a^).
7 r The quantity under the radical sign
in the denominator may be put under the form
(«+f-j^^('+i+J-+^)
= («+|Y + ,-^*:
/» C^O/
Making a; -f -^ = <, and g — ^ = ^^ ^® ^^^® ^^ = ^^' *^*^
/_-^, = /(< + V<M^^) by Ex. 3.
In this last, substituting for t and a their values, W6 have
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MISCELLANEOUS EXAMPLES,
dx
329
8. /Vaj' + px + s <^ =/ 1 ('^ +1) + 2 - "4- f <^^-
Put a; +1 = /, 2 - ^ = oS then
/Va!« +joa; + q dx = /V^' + o» cB
= ^ < V<* + a^ + -2 Ut + VP + a%
by Ex. 5. Replacing t and a' by their values, we have, finally,
/Va;" +iJX + q dx = -(x+^ Vx' +px + q
+
9.
/;
Ih'^'i:
^ + ^ + v5M-^
'^ + q\
dx
\/2ax — x^
and dx=^ — dt : therefore
Let x^a — t; then 2ax — x^ =:a* — t\
/dx ^ p d^ ^
s/2ax — a;2 "" "~ J V«2~=^*^ ""
COS."^-
a
.a?
= co8.~^ = vers.
a a
Puta; =
1-^
; then
r ax
yxs/'lax — a^
<:/a; :
ac?<
/j
dx
V2ax — a'
-J, and therefore
(1-0^ _1 /• rff
Vl - 1^
42
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330 INTEGRAL CALCULUS,
1 . _,, 1 . . x — a
= - sm. *^=i-8in.~*
a a X
11. fx COB. axdx. Assume u =: - , v =z ain, dx : whence
dv=:a cos, axdx and Jx cos, axdx =z Judv: therefore
/J a; sin. ax /»sin.aa3 ,
X COS. axdx = I ax
a J a
X sin. ax cos. ax
12. fe""^ sin, axdx. Put u = — - — , v = c""' :
•^ c
>, . - sin. ax ,^ /•oe'^'cos.ax ,
J e'^'^sm, axdx = ^ J ^'
But we have, in like manner,
rae^cos.aj, ^^ ^ acosax ^„ raHiuax^^^^^ .
J c c^ ^ J c^
hence
^ . , sin. ax ^^ a cos. ax ,^ raisin, ox ^^ ,
/ e^^sm.oxdx = e*^^ « — ^ "~ / 2 — c*^*cfe,
t' C C t/ c
which, by transposition and reduction, becomes
^ . , e'^'^fcsin.ax — acos.ax)
fe''^ sm. axax = — ^^ t—, — o -'
J a^ + c-
13. fe'^'' cos, axdx.
Proceeding with this as with the last example, we should
find
r , e*''(ccos.ax + asin.ax)
/ e"^^ cos. axdx = — ^^ r-j — 2 '
J a- -\- c^
14. /__i- ,=f ^^
J \/(q + px — xM «'
^hU-t)l
Fnt q+^ = a\ x— ^ = ^: .-. dx = dL
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MISCELLANEOUS EXAMPLES. 331
Therefore
dx /• (U _ • -1 ^
Making g + ''^ = a*, x — ^ = /, we have
■I 1 <j
/• <'a; „ 1 , , <^ ,
16. / — , , Put X = - : whence dx= — - . *i.i
•Z a;v x* — a* f <-
r dx _ /• .'/ __l-i"
1-1^ 1 - _ «
a ax
But, Bince
sin""^ — h cos.-^ - = -, Bin-'* -= 'x*.'' . '
a; X 1 a a a </. *x z
1 ;t
hence, throwing into the coiirUfci.t c=f y^jhr"<-* ^'^', *^
' a 2
may write
/cir 1 ,«
XV a? — a' a x
Digitized by VjOOQIC
330 INTEGRAL CALCULUS.
l..__.. 1
— - ain — I
sin.~*< = - sin."^
X — a
a a X
/x
X coa, axdx. Assume u= -, v = ain.ax; whence
a
dv = a COS. axdx and fx cos, axdx =1 fudv: therefore
/J a; sin. ax r&in.axy
X COS. axdx = / dx
a J a
X sin. ax cos, ax
H ^2 "
a
sin. ax
12. fe^^ sin. axdx. Put w = — '- — ; 17 = 6*^';
r . , sin.a.'c ^, rae"^' COS. ax ,
J e*""^ sin. axdx = e""^ — I ax.
But we have, in like manner,
hence
^ . , sin. ax ^^ a COS. ax ,^ /•a^sin.ox ^^ ,
fe'''sin,axdx = — - — e*^^ -^ — e^' — j ^2 — c*^*cfe,
which, by transposition and reduction, becomes
r . , e'^'^fcsin.ax — acos.Gtx)
/ e'^-^ sm. axax = — ^ ^— j — ^ ^•
13. Je'^^cos.axdx,
Proceeding with this as with the last example, we should
find
r , e *^'(c COS. ax-\-a sin. ax)
/ 6*"* COS. axax = — ^^ : , , .. ^*
•/ a- + c^
14. / - ^^ ,=/•- ^^
J k/(q + »x — XM «'
Jh^(^-I)l
Putg+^ = a2, ic-| = ^; .-. rfx=:rf^.
Digitized by VjOOQIC
MISCEILANEOUS EXAMPLES. 331
/dx __ r dt _ • -1 ^
= sm. ^ f_ __ sin.~* - ± .
15. /V(?+i>^-^^)rf^=/J W^-(a;-fJ|rfa;.
Making q+'^—zna^j x — ^ = t, we have
/\/(? +-P^— x'')dx=J\/d' — t'' dt
— _ Va'^ — ^' + IT sin "^ - ' by Ex. 2,
2 ^2 a ^ '
16. / — . ^ . Put x=i -I whence dx=: -, and
J x\x^ — a^ t V
rfx c dt 1 <?<
V^*^ — a*-* •^Vl — «'^''
'X^-^
= Bin. ^at^=. sm."^ - •
a ax
But, since
. .a .an \ , .x \ . a 1 tz
Bin~^- + cos.~^ -=^-y sm.-^ -=: cos.~^ -:
X X 2 a a a x a 2
1 7t
hence, throwing into the constant of integration, we
d u
may write
/dx 1 , a
—7 -, ^ = - COS.-* -.
x\ x^ — a^ a X
Digitized by VjOOQIC
334 INTEGRAL CALCULUS.
l,l + 8m.a;_,^^«-2^ + ''''-2'^
=u : ' =1
2 1 — sin. X
COS. -a? — sin. -a;
= Jtan.g + |
23. f^-^ = f gjg;'^ + «»«•''" dx
J sm. X COS. X J sin. a? cos. x
=y*(tan. a; + cot. x)dx
1 17- ysm.a; ,.
= — icos.a; + 6sm.a;=:6 =fctan.a;.
COS. a;
dx r sin.^ x + cos.^ x
24. /• . /^ ^ ^/-ElilJ
•/ sm.'* a? cos.'' a; J sm.^
dx
25. f-
= r(sec.^ X + cosec.^ aj)rfx'
=;: tan. X — cot. X,
dx
-f- 6 COS. X
dx
/ X x\ I X ilj\
a f sin.2 - + COS.' 2/ "*" ^ \'^ 2 "" ^^^'^ 2/
sec. ^-c?aj
X
a + 6 + (a — 6) tan.^ -
by observing that
.«a/, nX ^ n X , n X
8111.^ ^ + cos.-* = 17 COS. a; = cos.-* ^ — sm.^ - ,
and dividing the numerator and denominator of the result by
X
cos.^ ^. When a > 6, the last integral may be put under the
form
Digitized by VjOOQIC
MISCELLANEOUS EXAMPLES,
335
2 ■ d tan. ^
a — 2
When a <^bj we have
rfx 2
L-
d tan. -
2
+ b COS. a? 6 — alj_[_<^
, tan.2 -