what is the definite integral of (ln^(2)(2x))/(3x) with an upper limit or boundary of e/2 and a lower limit or boundary of 1/2?

Answer 1

If what you mean is ln 2^2x/3x ...

ln 2^2x = 2x ln 2, then you integrand for x≠0 is (2/3) ln 2 and the antiderivative is (2/3)xln 2

which you then evaluate between the given limits of integration.