1876.] Evaluation of a class of Definite Integrals. 



Now in the equation 

 sin 2ne = 2n cos 6 jsin d - '^-^ 2' sin'^ 



put 



2 COS ^ = ^ + - , 2ismd = x , 



and we obtain 



(1\ dx 

 X j — and in- 

 tegration between the limits and oo , gives (2), since 



= 2 f x^^-'cf) (x - i) dx, from (4), 



while the right-hand side assumes the form in (2) in virtue 



of (5). 



§ 3. The proof may be presented in a somewhat different 

 form as follows : — 



First, (/being an even function as before), show that 



fj{x-^yx=fj{x)dx (7). 



This is readily effected, for if 



(a positive), then -y- = — I fix ) — : transform this integral 



by taking x= - , and we have 



X 



du r" rt fa \ dx r" ,, / a\ dx 



f[--x] — = / h»-- — 



d/CC J ^ \l(/ / oC J p. \ M// 01/ 



