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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 examiples of the formula 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

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? (a? + 2 a').
J (– )
Let n = 6,
rdx

la 5ar 5.3a*x 5.3
6+ 64 + 6.4.2 6.4.2

(a

=-(

a'sin

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By this means the integral is reduced to
- da

ī = sec-1x when n is odd,
J x (x - 1)!
• da

- 1)
and to

- when n is even. J os (ava – 1)

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