Expert Reply
Mon March 12, 2012
In the 1st question let ex = t so we get ex dx = dt 
Integration of cos2t . dt can be done by using formula cos2t = (1+cos2t)/2 .
 
In the second question put 1/x = t , so -1/x2 . dx = dt that makes integration of -et .dt that is simple
 
In the 3rd question , add and subtract x in the numerator , we get numerator as (cosx + x) + x(sinx-1) that can be splitter to get 
 
(cosx+x)/denominator + x(sinx-1)/denominator = 1/x + (sinx-1)/(x + cosx)
 
The above expression can be easily integrated by substituting x + cosx = t
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