Combining the pairs of conjugate partial fractions according to equation (29), the first pair becomes and similarly will the other pairs of conjugate partial fractions be compounded; so that the following series will be formed, 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 Sevent a s log (e – 1) + gece log (20–22.00 +1) 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 of 0 S 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 (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 n-2 n-2 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 Ex. 1. Setia cos j V-1 sin, cos IN-1 sin 31 40 INTEGRATION OF RATIONAL FRACTIONS. [24. therefore the coefficient o–cos -v-1 sin –cos +/-1 sin ................; 2xcos ** - 2 371 , and Ex. 2. Se 1 / (2c cost – 2) dx 1 ; (20 cos en – 2) dx 22–2–008 +1 51 –2x cosm +1 le |