∫﹙1,е)sin(㏑x)dx寻答案和做题步骤

题目

∫﹙1,е)sin(㏑x)dx寻答案和做题步骤

答案

令t=lnx,则x=e^t,dx=(e^t)dt
I=∫ (1,e) sin(lnx) dx
=∫ (0,1) (sint)(e^t) dt
=∫ (0,1) (sint) d(e^t)
=(sint)(e^t)|(0,1)-∫ (0,1) (e^t)(cost) dt
=esin1-∫ (0,1) (cost) d(e^t)
=esin1-(cost)(e^t)|(0,1)+∫ (0,1)(e^t)d(cost)
=esin1-ecos1+1-∫ (0,1) (sint)(e^t) dt
∴2I=esin1-ecos1+1
I=(esin1-ecos1+1)/2