47.] If a = e = 0, the integral takes the form in which case we must assume as above; viz., if I is the least common multiple of the denominators q, s, = 6 23 22 - + - + -log (z+1)} z 7 5 Other methods of rationalizing irrational functions by means of substitution must be left to the ingenuity of the student. In many cases however when the irrational term is of one or other of the following forms we may rationalize the expression by the substitutions indicated; if there is involved 48.] Our object in the present Section is, by means of integration by parts, to make the integrals of certain irrational functions depend on those of similar forms and lower indices, and thereby to reduce them to forms whose integrals are fundamental or have been already determined. A careful examination of the process of reduction of the rational forms which were integrated by this method in Articles 27-30, and of one or two of the following formulæ, will give a clearer insight into the method than any general remarks and rules, and therefore I proceed at once to examples. By means of which the integral will at last depend on, The formula, it will be observed, is always applicable when n is odd; but if n is even, ultimately, when n = 0, it becomes infinite, and fails to give a determinate result. x" dx (2ax-x2) .nf. x" dx (2 ax-x2) v = (2ax-x2); = −x"−1(2ax— x2)3 + (n − 1) f x*−2 (2 ax —x2)31 dx · · -1 — — x* ̄1 (2 ax − x2) 1 + (n − 1) x" dx (2ax-x2) (2 ax — x2) x" dx (2 ax-x2) — } ax—x2) xTM"−2 dx (2 ax — x2) "-1dx (2 ax − x2) (2 ax − x2) 1 + a (2n−1 + a(2n−1) f xn-1dx (2 ax — x2)} x"-1dx (2 ax-x2) x"-1dx + (92) n (2 ax-x2) ; By which process the integral will at last depend on By a process exactly similar to that of the last article it may 53.] Integration of (a2-x2) dx, where n is odd. · · [(aa—x2)3 dx = x (a2 + x2)§ +n [(a2 —x2)4−1‚x2 dx 小 = x (a2 — x2)2 + n | (a2 — x2)1—1 { a2 — (a2 — x2) } dx = x (a2 — x2)" + na2 ((a2 — x2)3-1 dx — nf (a2 — x2)3 dæ ; (n+1)f(a2—x2)2dx = x(a2 —x2)3 +na2 [(a2 —x2)3-1dx ; .. By which means the process of reduction may be continued until |