(40) By the same assumption we find that

(43) a In like manner by assuming x = ≈ {(1 + x®)‡ − x2}}, we find

These transformations are taken from Euler, Calc. Int Vol. IV. Sup. I.

CHAPTER II.

INTEGRATION BY SUCCESSIVE REDUCTION.

THE method of integration by successive reduction is applicable to a great number of functions, and is the process which in practice is generally the most convenient. I shall here only give the principal formulæ of reduction with a few examples of each, taken chiefly from those integrals which more commonly occur in analysis. The reader who wishes for more numerous examples of the formulæ is referred to the Integral Tables compiled by Meyer Hirsch, from which work a great number of the examples in this and the preceding Chapter have been taken.

Ex. (1) Let the function to be integrated be

By means of this the integral is made to depend on

(a2 + x2)n

x

2n-3 1

+

dx

(a2 + x2)* −1

2n−2 a2 (a2+x2)"−1 2n-2 a2

By this the integral is reduced to

(5) Let the function be (a2 - a2), n being odd;

duction is the same, excepting that both terms are negative. The final integrals to which it is reduced are