32 DIFFERENTIAL AND INTEGRAL CALCULUS. 



so that f 1 * Bin m axfe=^ i (^ am 



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Putting i TT # for x in this result, we get 



/ cos w ic^ic = - \ w^ 



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and we also see that I sin m aj^r= j coa m xdx, (as is 



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obvious, since the elements of each are exactly the same, 

 but in reverse order). 



' So also I sin n xdx = | 8iu"xdx + I sin n a5dkc, and 



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if in the latter term we put TT x for x it becomes 



/"a 71 " . /" 7r . /"Jr 



I sin" #<&?; whence ; sin" xdx 2 I sin n x^ic ; (as is 



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obvious from first principles, since the elements of the 

 integral from TT to TT, are the same as those from to |TT 

 in the reverse order). The formula? of reduction I., II., 

 III., IV. can be all deduced from the formula for fain" xdx 

 by putting x = a sin z in I. and III., and x = 2a sin 2 #, in 

 II. and IV. 



A very useful result in finding the values of definite 

 integrals is the following : 



f(j>(x) dx = I (f>(a x)dxj which is obtained at once by 

 ^0 



putting a x for x in the first member. It must be 

 remembered that a definite integral is not a function of x 

 at all, and that it is of absolutely no matter by what 

 symbol we denote the variable in the subject operated 

 upon. Thus we might get for the above 



f <t>(x)dx={ <f>(a-z)(-dz), 



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since when x = a, a x or z 0, and when x = 0, z = # , 

 = f 4(a-z)ck, 



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but this is exactly the same thing as I <p(a x) dx. 



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