An Elementary Treatise On the Integral Calculus: Containing Applications to ...
An Elementary Treatise On the Integral Calculus: Containing Applications to ...
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When m and n are both fractional, the preceding is one of the most important definite integrals in analysis.
We purpose in a subsequent part of the chapter to give an investigation of some of its simplest properties.
Examples.
!1 Sl 2, {\*(i'x)idx.
1^ dx. Am, • 3.7. 11.13 5.7.9- U- '7 Digitized by Google Elementary Exampks. 1 1 7 93. Values of srn^xdx and ooB*^xdx.
One of the simplest and most useful applications of definite integration is to the case of the circular integrals considered in the... commencement of Chapter IIL We begin with the simple case of 1 sm'^xdx.
If in the equation, (Art. 56), f • « JF cosiCsin*^^ii: w- i f . ^ .
J n n } ve take o and — for limits, the term vanishes 2 n for both limits, and we have sin^^xdx = sm^^xdx.
Jo n Jo Now if n be an integer the definite integral can be easily obtained ; its form, however, depends on whether the index n is even or odd.
(i). Suppose the index even, and represented by 2;w, then sin*"* a? dx = sin'"*"*aj dx.
Jo 2m Jo Similarly sm^^'^xdx = -\ Bia^"^xdx; Jo 2;;^-2jo «nd by successive applications of the formula, we get fsin-^efe = i^3^ ----(^'"-0 .
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