Oscillating Di ri chiefs Integrals. g^7 



I Z, (/) = /' V.t) cos a(x) ^^^ dx, L (/) = f '(>{x) sin a{x) -^^^ cZa-, 



7. 



r^l/) = / ('{x) cos ff (a-) '^ dx, IM) = / (>{x) sin a(x) ^"^ ^'^^ tZa-. 

 ./ a:" ./ a:; 



All these integrals are convergent if 



(35) ' x'<f>{x) <xa'{x). 



Hence we shall suppose that this is satisfied. Then, if we put 



{j,{k) =^'JA^lcos {?.x + <T[x)}dx, J,{1) =l'JPi^cos {äx-<j(x)} dx, 



\j,(/:) = I 'J^ sin [ax + rr{x)} dx, J,(Â) = f '^^ sin [/.x-afx)] dx, 

 \ ./ I) X ./ X 



we have 

 and 



f c(/) = i[/,(;o + j,(À) + /{/,(/) - j,(?.)}i 



(36) { 



I ^ (/) = i[ J, (/) + J, (/) - / {J, (;o - J-, (/)} ]. 



21- Integrah J.diul Ju At first we shall consider the integral 



JJÀ) = f'A^lcoäydx, 

 -'^ ^ Jo X -^ 



where y = /^x—o{x). As x increases from to ^. // increases from 

 — GO to '^/—'^^^-^{^), and Jy tends to infinity with /. Also 



f){x) 



./ -xa:{/ — (T'(a:)] ■' 



dy. 



Let s^ = ^/4^' 



then, l>y H-lemma 32 and Lemma 2. we have to separate the 

 following cases. 



