Question

高数定积分问题，求解

Answer 1

如图

Answer 2

解：换元法
令t=lnx.
x=e^t
dx=e^tdt
原是=积分sinte^tdt
=-积分e^tdcost
=-(e^tcost-积分costde^t)
=-e^tcost+积分coste^tdt
=-e^tcost+积分costde^t
=-e^tcost+coste^t-积分e^tdcost
=-e^tcost+coste^t-积分e^t(-sint)dt
=-e^tcost+

解：原是=积分sinte^tdt
=积分sintde^t
=sinte^t-积分e^tdsint
=sinte^t-积分e^tcostdt
=sinte^t-积分costde^t
=sinte^t-(coste^t-积分e^tdcost)
=sinte^t-coste^t+积分e^t(-sint)dt
=e^t(sint-cost)-积分e^tsintdt
令积分sinte^tdt=a
a=e^t(sint-cost)-a
2a=e^t(sint-cost)
a=e^t(sint-cost)/2+C。