Reduction formulas — for reference

These expressions can always be integrated if (m + 1 )/n be a positive integer. Four cases present themselves according as m, or p are positive or negative. 

m must be the positive, otherwise the index will increase, instead of diminish, by a repeated application of the formula. When m is negative, it can be shown that the integral (1) is equal to 

Formulae B, C, D, E have been deduced from (1), page 208, by the method of integration by parts. Perhaps the student can do this for himself. The reader will notice that formula B decreases (algebraically) the exponent of the monomial factor from m to m - n + 1, while C increases the exponent of the same factor from m to m +1. Formula D decreases the exponent of the binomial factor from p iop - 1, while E increases the exponent of the binomial factor from p to p + 1. B and D fail when np + m + 1 = 0 C fails when m + 1 = 0 E fails when p + 1 = 0. When B, C, and D fail use F, page 204 if E fails p = - 1 and the preceding methods apply. 

If (m + l)Jn he a positive integer, the expression in brackets can be expanded by the binomial theorem, and integrated in the usual manner. By comparing (2) with (3), it is easy to see that (2) can be integrated when m+1 j(n-l) + l 1 1 

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