if we expand the sinx,logx terms when divide byx we integrate otherwise we cannot why?

please reply to my answer.

sinx/x cannot be directly integrated in the for min which it is given .but if we expansion series of sinx and divide that with x then we can integrate but that is also not finite .it is infinite expansion.

can anyone explain this  in a detailed way.

please asnwer my question. urgent.

please give me the proof.