f(t)=1/T⌠1/t (1/x)ln(tx) dx

Question

If

     f(t)=     ⌠^1/t (1/x)ln(tx) dx                      

            _1/T⌡

where T is a constant with the same physical dimensions at t, find f'(t). Try to write the answer in a form which makes dimensional sense and which best exhibit the sign of the t>T

 

 

 

The equation above is and integral the top is 1/t and bottom is 1/T.

Answer 1

Let u = 1/t. then f'(u) = (1/u)(ln u) times the derivative of u...chain rule must be applied to the upper limit variable.

simplified, this gives f'(t) = t ln(1/t) (-1t^-2)   and you can further simplify this.