EVALUATION OF CERTAIN DEFINITE INTEGRALS. 145 



A* 



= r tan" 1 --fa log \/(a 2 + ?* 2 ) + C, 



and generally 



sin 



(2), 



where F l (ra) does not become infinite for a = 0. 



Replace r by every quantity represented in (1) by the 

 general symbol a. Multiply each result by the corresponding 

 coefficient A, and add. 



Then, in virtue of the conditions, 



2L4a m ~* = 0, ^AcT'* = 0, &c. 

 we shall have 



'*-** frS] tanJ I + a * AF > ( a )' 



Put a = 0, then tan" 1 - = - , according to whether a is 

 > or < than zero. 

 Thus we have 



From hence, the truth of our theorem is obvious. 



Of the ambiguous signs outside the symbol of summation, 

 the upper is to be taken when n is of the forms 4p, or 4p + l. 



When a is positive, we must take the upper of the am- 

 biguous signs under the 2. 



It will be remarked, that in obtaining (3) we have elimi- 

 nated all the constants at once, instead of getting rid of them 

 one by one by particular conditions at each successive inte- 

 gration, and that the generality of this method enables us to 

 recognize a class of definite integrals, which are all deduced 



-00 



from the known value of I e~ a * cos rx dx. 



J o 



10 



