Please find integration of (x).( e-x) w.r.t. x between limits 1 and 3 from ab initio (NOT BY DIRECT INTEGRATION). Consider partitioning of span [1, 3] into n small divisions of h such that nh = 3 - 1 = 2. Now find value of above integration by summing up area under each division and taking limit h tends to zero. i.e. plz evaluate
I = Lim x tends to zero (h).[ (1).( e-1) + (1+h).( e-(1+h)) + ------------------- +(3-h).( e-x(3-h)) +​(3).( e-3)]

Since the integral as a sum of limits can be written as

............(1)

 

here we have to find the 

here a=1 and b=3,  ,i.e nh=2,

 

........................(*)

 

 

.............(2)

 

...............(3)

 

let 

 ................(4)

therefore 

thus 

 

 

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