If `int_0^1(e^t)/(1+t)dt=a ,` then find the value of `int_0^1(e^t)/((1+t)^2)dt` in terms of `a` .

Answer

`a+1-e/2`

Explanation

`a=int_(0)^(1)(e^(t))/(1+t)dt=(1/((1+t))e^(t))_(0)^(1)=int_(0)^(1)(e^(t))/((1+t)^(2))dt`
(Integrating by parts)
`=e/2-1+int_(0)^(1)(e^(t))/((1+t)^(2)) dt`
or `int_(0)^(1) (e^(t))/((1+t)^(2))dt=a+1-e/2`