Evaluating lim x → 1 ∫ 1 x ln 2 ( t ) d...
Answer & Explanation
In terms of the evaluation of the limit: applying L'Hôpital's rule once forms
where the chain rule implies that the derivative of the denominator is
From here you can apply L'Hôpital's rule two more times to find
To answer your second question, the result follows from the fundamental theorem of calculus. Let f be a continuous, real-valued function defined on the closed interval [a,b]. Suppose F is defined for all by
Note that we could change the lower integration constant to any other positive real constant (as the fundamental theorem of calculus would still hold) to deduce
However, if the upper limit of integration was instead of x, then we would find