An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

194 The integral of rational algebraic functions. Bk. IIl. ch. II.
Vi- la-+(2+) +2 7+.. k+2J-})
x = -- eai n =e n COS — o + isin.
k-==, 1, 2,... n- 1.
Accordingly the type of the constant numerator of its partial fractions is:
1 (m - n) + - 1 (2 k +1)-
Ak e Me C n
nT)b e n n
therefore:
Jx -dx (2kh+1') ( i( + A
+ 5 +e ne(i n —)~- a
If we separate the logarithm of the complex quantity (~ 74) into its
real and imaginary parts, supposing x real, and call:
Pk --- 1 IX -- 2xcos - 2,
sin ll- 210- 1
Q, tan-1 it____.o+2k+li
a +t 2k + 1 7t
COs - - X
incorporating the multiples of zri in the arbitrary constant we may
write this indefinite integral:
X 1, (S ( +1)2 n
x__ -f _ _ (P + i Qk) + C.
Let the definite integral be required between the limits zero and cc,
when all roots of the denominator are complex or negative, thus
- z < ca < + 7r. To obtain it, Pk and Qk have to be determined
at the limits 0 and o: they must however remain continuous in the
entire range from x = 0 to x = o.
The value of each P vanishes for x. 0 and continuously increases as the values of x increase. But inasmuch as
X2 > 2x cos + 2+ -- 1 I,
if we write:
a + 2k + 17r
/ 2 a cos - - 1
Pk = (x2) + 1 1 - 9 -,
and combine all terms multiplied by - l(x2) in the sum, they amount to
7 - -n — 1 2 n n --- m i
I (x2) =e(2+ ') n - (x) * e — 0;
k=- 1- e n
this formula is no longer valid when m = n. In this case the integral