Evaluate a integral by taylor series

http://www.hkedcity.net/sch_files/a/kst/kst-06185/public_html/esinln.jpg correct your answer in 4 decimal places!! ============================== the first step in my mind is find out the taylor series of e^x,sinx and lnx, and then multiply them all to obtain a new polynomial approximation so that we can... show more http://www.hkedcity.net/sch_files/a/kst/kst-06185/public_html/esinln.jpg

correct your answer in 4 decimal places!!
==============================

the first step in my mind is find out the taylor series of e^x,sinx and lnx, and then multiply them all to obtain a new polynomial approximation so that we can integrate it without difficulties.

but the questions are....

1. although i can find the taylor series of e^x,sinx and lnx easily, how can we ensure the accuracy (4 deci. places)?
Update: 2. 乘曬e^x,sinx and lnx之後,will the accuracy change?

3assume that i can solve the above 2 problem, will the accuracy change AFTER the integration? and how?

i think it is difficult for me, plz give me some hints or reference to tackle this problem.
thz
Update 2: hope that you won't answer me that...

you can substitute 1/2 and 1 to check the accuracy.
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