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Combining the pairs of conjugate partial fractions according to equation (29), the first pair becomes

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and similarly will the other pairs of conjugate partial fractions be compounded; so that the following series will be formed,

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If n is even, then equation (48), by means of Art. 64, Vol. I, becomes 1 1sı 2 x cos —

2x cos i - 2

+ ... 22 - 2 x cos

+ 1

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Sevent a s log (e – 1) + gece log (20–22.00 +1)

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re cOS

COS

2

I dx 24.] To determine

J 2+1 Let n be even ; then, in Art. 65, Vol. I, it is proved that the roots of x*+1 are cos+v=Isin, cos me? V īsin ... cos ^»? + V=1 sin "72.

Now I am anti- n -, since w" =-1 for all the roots of x" +1 = 0; therefore the coefficient of - 1 is 2008" + —I sin };

#—cos .-v-1 sin of —1 i s - {cos"-v1 sin};

x-cos+ V–1 sin

. . . . . . . . . . . . . . . . . .; and combining the pairs of conjugate partial fractions, according to equation (29), the first pair is

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of

0

S

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SI

. 39 and the other pairs give similar results ; so that de 1/(2005 –2) de _ 1 5/2009 95 – 2) der

x2–2xcos +1 n) 22–2.2008 + 1 .... ? (20008" — 2) des

23–2æ cos"=17+1'

OS

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(53)

each of which must be integrated according to the process indicated in the preceding article.

Again, let n be odd: then the roots of 2*+1 = 0 are

00. v-1 sin, come tv-Tsina

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n-2

n-2
cos — +N -1 sin T, -
" -

1;

n so that if the conjugate partial fractions are combined by a process similar to that employed when n is even, the last pair becomes

n-2 Os

- 2

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Ex. 1. Setia
The roots of 3+1 = 0 are

cos j V-1 sin, cos IN-1 sin

31

40

INTEGRATION OF RATIONAL FRACTIONS.

[24.

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therefore the coefficient
of — l i s - 1 (cos + v–I sin );

o–cos -v-1 sin
of — 1 i s - (cos" - V=Isin);

–cos +/-1 sin

................;
and the pairs of conjugate partial fractions compound into
12xcos – 2

2xcos ** - 2
* x2–2x cos + 1 * 20 –20 cose + 1

371

, and

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Ex. 2. Se

1 / (2c cost – 2) dx 1 ; (20 cos en – 2) dx

22–2–008 +1 51 –2x cosm +1

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