Jaden and his family are using a map to drive down to Napa Valley. On the map, the distance is given by the equation c = 1.3m, w
here c is the number of centimeters on the map and m is the corresponding real-world distance in miles. If they cover a distance of 18 miles, that would mean a distance of how many centimeters on the map.
2 answers:
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Replace x with π/2 - x to get the equivalent integral
but the integrand is even, so this is really just
Substitute x = 1/2 arccot(u/2), which transforms the integral to
There are lots of ways to compute this. What I did was to consider the complex contour integral
where γ is a semicircle in the complex plane with its diameter joining (-R, 0) and (R, 0) on the real axis. A bound for the integral over the arc of the circle is estimated to be
which vanishes as R goes to ∞. Then by the residue theorem, we have in the limit
and it follows that
