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n-1 and consequently fë sin" r=-- cos x sina
x (n-2) (n-1) (n-3)
1 Jr sin- x = cos x sini n (1-2) n-1
(n-1) (n –3) cos r sin
cos x sin n (1-2)
(1-2) (1-4) (n-1) (n-3) (-5)
cos x sin"-71-&c....1 (1-2) (n<4) (6) (n-1) (143) ... 2
cos x; a formula which is true only when n n (1-2) (
14) ... 4 is an odd number, and then the integral depends simply on the quantities cos x, sin x. But when n is even, instead of the last term
(n-1) (143) ... 1 cos of the series, which would be of the form-
..O sin (n-1) (n—3)... 1 yö sin*-* x=+
(n-1) (n-3)...I we shall have + 2.4...(n-2) =
2.4....n Consequently in this case the integral will be 1
(n-1) (91-3)...1 cos x sin"-1
cos x sino-3.1--&c.... + n (1-2)
4.2 Examples. Së sin x=-} cos x sin* cos x sin2
COS I; 5.3
5.3 and få sin x=- sin' r cos
sinI cos x
sin ICOS 6.4
6.4.2 5.3.1 2.4.6 88. Make r=90°—2, and we shall have :=-2, and sin r=cos z
n1 and si cosa z = sin cosm-12 +
sin % cos z +
n (1-2) (n-1)(n-3) sin 2 cogn–5 % +
(n-1) (n—3) (1-5) sinz cos; n (n-2) (n<4)
1 (n-2) (4) (146) (n-1) (n=3) (2–5)...2 + +
sin if n is ndd ; and if even, n (n—2) (1-4)...I the last term will be (n-1) (n-3) ... 1
4 For example, sy cos' y=} sin y cost y + sin y cos'y +
4.2 siny; P
sin oy cos 'y +
y sin +2
5.3 and sy cos oy=; sin y cos sy +
sin y cos Sy+
sin y cos y + 6.4
89. Let it now be required to integrate y sin y cos "y; since flux (sin py cos 'y) =
y sin' yy-9 0092-1 y ein P+1 y y, we have jý sinp-' y cos?
y. P Therefore Sý sin y cos "y=
y cos'y+ m+1
m+1 Sý cosa?
y. Substituting 1-cos'y in place of sin "y, and transposing, we have Sý sin my cos "y=- sin
y cos y +
1 - Sý sin"
y y+ m+n m +12
y, if n is odd; or if n is even the last (m + 1) (m+1-2)...m +1 term will be (11-1) (n-3) ... 1
Jy sin" y. (m+n) (m +242) ... m +2° 90. Make y=90°—2, we shall have si cosa sin"
m+n sin-1 x cosm +1
(n-1) sina-5 % Cos'
(m+n) (m+n-2) (n-1)(n--3) com +1 z sin62
(n-1) (n-3)...2 cosm+1 (m+n) (m +16—2) (m +1-4)
(m+n)(m+n-R)...m +1 if n is odd, or to the term +(n-1) (n—3)...l fż.cosa 2 n
if n is even. (m+n) (m+n-2)...(2+2) For example the first formula gives lý cos y sin' y=j sino y (cos? y+})=sin" y 1-sin’ y), and the second Sý cos y sin y=-- cos 4y (sin *y+ * sin ’y+}). Hence these two results ought to be equal, or at least to differ only by a constant
1 quantity. In the present case, this quar.tity is
as we skall find
24 find by redncing the whole to sines, and comparing the two results.
2 sin" +1
y sin y
91. Let us now consider those fractions in which sines enter ;
j cos y y sin y; we and as the most simple are
у shall begin by integrating these. y
flux cos y The first
1.flur cos y
1-cos y 1+cos y
1-cosy=1 tan\ y by (art. 1.
1+cos y 20 Trig.)=1 tan ) y.
To integrate the second, let y=90°—2, and we shall have y=-1, and sin y=cos z; therefore 2
- tan (45°—1)=-l cot (45° +1 z)=l tan (45° + 2).
The third fraction sy.cos y has for its integral
r_flux sin y=l sin y=fy cot y.
l The fourth sy sin y=-l cos y = 1 sec y = jj tan y ; similarly
cos y 2y
=l tan y. sin 'y
= усов у 92. This premised, let us investigate the integral of the formula y We have already seen (87) that Ký sin" y=
cos y sin
-fy sin?-? y; therefore making n–2=-, or n=2–m we
shall have s
cos y sin?-m
y sin" y m-1
y 1 cos y m2
(m-2) (m-4) y (m-1)
(-3) sin" y (m-1)/(m-3)(m-5) sinas y ...to the term + (m- -2).(m- -4)...1
.; that is, to (n-1) (7-5)...2'sin y
cos Y if m is even.
(m-2) (M-4) ...1
I tan y, if m is odd, and to
(m-1) (m--3)...1 93. Suppose y = 90°—, and the preceding formula will give
cos" z (m-1) (m-3) cos_3
+ &c....to the term (m-1)(m-3) (m_-5) cosm (m—2) (m—4)...2. sin 2, if mis even, and to the term (m-1) (m
(m—3)...1 (m—2) (m-2)... 1
I tang (45° +12), (m-1) (m—3)...2
Itan(45° + y). coso y ' 6.4 cost y ' 6.4.2 cos” y ' 6.4.2 Hence it is easy to integrate the formula y cosa y, for if m is an
gin Y +
(1-sino 3)", which is evidently integrable for any value of n. If m is any even number 2k, then y cos" y_j (1—siny)*
which, when developed. is easily integrated by the formula for /_y
95. The same process will also apply to y sin“ y
; and the formula
cos" y y
may be integrated upon similar principles ; so that it is sino y cosa y easy to integrate any fluxional expression containing sines and cosines, provided that they are susceptible of integration.
Examples. 1. Required the fluent of 2 sin* z cos?z?
sino x cos z?
ON THE INTEGRATION OF FLUXIONS CONTAINING SEVERAL VARIABLE
96. In any function t, of two variables x and y, if we first take the fuxion upon the supposition that x alone is variable, every term not originally containing some power of x will disappear, and the result will be of the form Pr. If we then take the fluxion of Pi, upon the supposition that y alone is variable, the result will be of the form P'xy, and by the double operation, all terms not originally containing both x and y, will have disappeared.
Again, if we take the fluxion of the same function i, upon the supposition that y alone is variable, the result will be of the form Qy; and if we then take the fluxion of Qý, upon the supposition that : alone is variable, the result will be of the form Qyr, all terms being excluded here also which did not originally contain some powers of both x and y.
Now in any term including powers of both x and y, we shall obviously obtain the same result, whether we take its fluxion, first supposing only x to vary, and then take the fluxion of the result, supposing y alone to vary; or whether we first suppose y alone to vary, y
y and then x. Hence we shall always have Pixy=Q'yx, or more simply P=Q.
The quantities På, Qy are called the partial fluxions of the function t, the first taken relative to x, the second relative to y; and P Q are called the partial fluxional coefficients.
Example. Let the function be z* + xy +a' y. Then the partial fluxion relative x is (4x} + 3x? y) x; and the partial fluxion of (4.x} + 3.22 y) e taken relative to y,
is 3.x xy. Again the partial fluxion of 3+ + x?y + a’y? relative to y (r +2 a* y) y, and the fiuxion of (x2+2a2 y) y, supposing x only to be variable, is 3x? y x, the same result ag before. The partial fluxion of the function t,; taken relative to x, that is r ,
w; and the par
supposing alone to vary, is usually expressed by 3;