The Rev. Brice Bronwin on some Definite Integrals. 491 



giving a general direction to the main channel, and used for 

 roads, or otherwise, until the most favourable time of tide, 

 then to be thrown into the main stream, so that the volume of 

 ebb waters may even be larger at that time than prior to any 

 embanking on the river. 



These are, however, subjects requiring great local care and 

 study, and are merely mentioned to show that artificial im- 

 provements could readily be made in estuaries without dimi- 

 nishing the effective action of tidal waters, if these be so ma- 

 naged, and the general arrangement such, that their useful 

 powers be not deteriorated. 



June 10th, 1843. 



LXIX. On some Definite Integrals. 

 By the Rev. Brice Bronwin*. 



r r , HIS paper contains an easy method of obtaining some de- 

 A finite integrals. They are taken between the limits x = 

 and x= oo ; m and n are positive integers; P (n) = 1 . 2 . 3...w; 

 and Az = 2 ; also i = 0, 1, 2, &c. 



/*. fsmx\ m , x „. , 

 Let y z m =J d x \—^r) cos \ z x ) J { x ) 5 



then Ti< = -*J d *\—) cos ( z + 1 )^/W 



- \f dx Cir)" cos {z - 1)a?/w = t A -^~-\ ; 



therefore y z = ~ Afdzy 2 - 1 , = — & m f™ d z m y z ~ m \ 



or 



fdxi J cos{zx)f{x) = ~A m J d z m /dxcos(z —m)xf(x). 



Differentiating 2 i times for 2, and making n = m — 2 i, there 

 results 



f—p\nx) m cos{zx)f(x) = ~^ A n lf n d z n Jdxcos(z - ot)^*). (A.) 



The integrations relative to % will introduce the corrections 

 C (z - mf- 1 + C 1 {z- m) n ~ 2 + &c, where C v C 2 , &c. will 

 be infinite; but it will be easy to perceive that A w_1 will 

 render them finite, and therefore A m will make them zero. 



Makey(.r) = c~ iX ; then J dxc~ tx cosrx = -% ^, 



* Communicated by the Author. 



