Concept of Definite Integral

Question

If !!int_0^1 (e^t)/(1+t) dt = e/2!!, then!!int_0^1 (e^t)/(1+t)^2 dt= ?!!

(a) !!0!!

(b) !!e/2!!

(c) !!e!!

(d) None

Solution:

Given that:

!!int_0^1 (e^t)/(1+t) dt = e/2 .....(1)!!

!!I= int_0^1 (e^t)/(1+t)^2 dt!!

Use integration by parts,

!!=> e^t(-1/(1+t))|_0^1- int_0^t (e^t)(-1/(1+t))dt!!

From equation (1), we get

!!=> I=-e/2+1+e/2!!

!! I=1!!

Therefore, option (d) is correct.

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