Oscillating- Diriehlofs Int-e^'rals. 45 



Hence, by (42), we have 



T,_ p{d) f /•^("^^H-'C''-) cosy , ^ ,,.1 



Similarly we obtain 



ru_ m / /■^'^^--)^-C'>-0 cosy , ^ ,.A 



Now /(^ + e) + a{d + e) = ß -\- is-V"(^ + e,), 



where < e^ < e, so tliat a"{d + e,) ^ o"{d) ; 



and, by (38), t-a"{d) > 1. 



Hence we have 



/•A(«+e),.(no cosy , f^ cosy , 



A v(^'-'^=.A v(F^'^^^^^^- 



Therefore we obtain 



Similarly 



Hence, in both of the cases (i) and (ii), we obtain 



26- Integral J'/^\ Finally we consider the integral 



Ju X '^ J x=a. x[X + a'{x)] "^ -^ 



In this integral, ^o^^y//^)-) i^ a steadily increasing function of x 

 in the interval a<a;<f. Hence, by the Second Mean Value 

 Theorem, we obtain 



