Answer
The solutions are:
a) $g(4,1) = -2\ln 2.$
b) $g(6,3) = -\ln 2.$
c) $g(2,5) =\ln \frac{5}{2}.$
d) $g(\frac{1}{2},7) = \ln 14.$
Work Step by Step
We will first calculate the integral and express $g$ explicitly as a function of $x$ and $y$:
$$g(x,y) = \int_x^y \frac{1}{t}dt=\left. \ln |t| \right|_x^y = \ln |y| -\ln |x| =\ln \left|\frac{y}{x}\right|.$$
Now we will substitute for $x$ the first coordinate and for $y$ the second coordinate from the given point
a) $g(4,1) = \ln\left|\frac{1}{4}\right| = \ln\frac{1}{4} = \ln 2^{-2} = -2\ln 2.$
b) $g(6,3) = \ln\left|\frac{3}{6}\right| = \ln \frac{1}{2} = \ln 2^{-1} = -\ln 2.$
c) $g(2,5) = \ln\left|\frac{5}{2}\right|=\ln \frac{5}{2}.$
d) $g(\frac{1}{2},7) = \ln\left|\frac{7}{\frac{1}{2}}\right| = \ln 14.$
