SECT. II.] EVALUATION OF AN INTEGRAL. 373 



375. The relation in which these two particular values are to 

 each other is discovered when we evaluate the integral 1 



/ 



I 



J 



dn e ^t cos nx. 



To effect the integration, we might develope the factor cos nx 

 and integrate with respect to n. We thus obtain a series which 

 represents a known development; but the result may be derived 



more easily from the following analysis. The integral I dn e~ n * cos nx 



is transformed to I dp e~^ 2 cos 2pu, by assuming r?t =p 2 and nx = 2pu. 

 We thus have 



/foo 1 /+&amp;gt; -J. 



I dn e~ nH cos nx = ^l dp e~& cos 2pu. A 



J -oo *JtJ - /r ^S 



We shall now write ~S 



Idpe~^cos2pu = ^ Idpe-^+fyu^- 1 + \ f&amp;lt;#p e--p a - 



~ u * Idpe^- 

 -- u * (dp e - 



V 



Now each of the integrals which enter into these two terms is 

 equal to A/TT. We have in fact in general 



and consequently 



= I 



J -00 



whatever be the constant b. We find then on making 

 b = T M s/^T, I ^ e&quot; 9 cos 2#w = e~ tt V^ 



hence I dn e~ nH cos nx = - ^ , 



j -oo *y^ 



1 The value is obtained by a different method in Todhunter s Integral Calcuhu, 

 375. [A. F.] 



