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The roots of the denominator are those given in Art. 23; and
since v(x) xm _ xm + 1 _ xm + 1
f(x) ~~ nx"~^ ~ nx" n'
we may determine without difficulty the coefficients of the several
partial fractions, and the form which a pair of conjugate partial
fractions assumes when they are combined.
„ . ., . , , xmdx , F(x)dx
By a similar process may we integrate ——r, and — ,
30 ~\~ J. OC ~\~ 1
when r(x) is integral and rational and of not more than (n — 1) dimensions.
To the general forms of Arts. 23 and 2-4 may also be reduced xmdx
a + bx"'
for if we replace bx" by azH, the integral becomes
26.] The process however for obtaining the integrals of many infinitesimal-elements of the preceding forms may oftentimes be much simplified by a judicious substitution; the selection of which must be left to the ingenuity of the student, because no general rules can be given. A careful study of the following cases will probably indicate the course whereby he may be led to such a simplification.
Price, Vol. n. G
x dx 11 d.x3
- 3 J x} (a + 203)
1 dz which, if 203 = 2, becomes a
°3 ) z(a+z)'
s; and the integral may be determined by the method of Art. 19.
Ex. 1. Srce
and which last integral may be found by the method of Art. 21.
27.] The following integrals might be determined by one or other of the preceding methods : but the process of integration by parts leads to a result more convenient, and better suited in most cases for finding the definite integral.
Integration of the
" (x2 + a)"
Now, as the integral in the last term of the right-hand member of the equation is of the same form as the original integral, but has the index of its denominator less by unity, so may the same process be repeated successively until finally » = 1, in which case the formula fails to give a determinate result : but the
s dx integral becomes 1 oz, and we have, see Art. 14,
The method is known by the name of Integration by Successive Reduction.
dx Ex. 1. Integration of a
(2x2 + a2)3
x dx let dv =
U = xm-1; (222 + a?)"? .. v = - V ini, du = (m-1)-2 dx ; r om dx - xm-1
m-1 2m-2 dx **)(x2 + a2)» = 2(n-1)(x2 + a2)n-1t2n-2)(x2 + a2)"-1: (59)
By which means the integration of the original element is made to depend on that of another element of precisely the same form, but whose numerator and denominator are of lower dimensions; and thus, by successive and similar reductions, the integral will be brought either into a fundamental form or to that of Art. 27; the formula, it will be observed, failing when n = 1. To f x2 dx
na: here m = 2, n = 4.
and the latter integral has been determined in Art. 27, so that it is unnecessary to repeat it.
A process exactly parallel to that of the last article gives a” dx
m-1 [ 2m-2 dx J (as?)* = 2 (n-1)(a? — 22)"-1 2n-2 J (a? —x2)-1:
1: (60) By which means the indices in both numerator and denominator are diminished, though the form is unchanged : and by a similar reduction we shall arrive at an integral either of a fundamental form, or of that of Art. 28; the formula fails, when n = 1.
s 25 dx
r x3 dx
SECTION 3.-Integration of Irrational Algebraical Functions.