#
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

**
**