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The integral of secant has an interesting history - Was just thinking about it and thought I'd share: @stevenstrogatz @mathyawp @maths_kath
int sec(x) dx = ln|sec(x) + tan(x)| +C
int sec(x) dx = ln|sec(x) + tan(x)| +C
In the Mercator projection Northwest, or indeed any other compass direction is a straight line. Creating a projection that satisfies this condition required the integral of secant.
Before Newton & Leibniz had done their thing.
Before Newton & Leibniz had done their thing.
It was initially done numerically by Riemann integration. The analytic formula was discovered because back then people had tables. Lots of tables. No calculators. So someone noticed that the table for the integral of sec(x) and ln|tan(theta/2)| were related.
Thus emerged one of the great open mathematical questions of the 17th century.
Several people proved it. One of the techniques turns out to involve partial fractions. The first use of partial fractions to solve an integral. Again, before Newton published the Principia.
Several people proved it. One of the techniques turns out to involve partial fractions. The first use of partial fractions to solve an integral. Again, before Newton published the Principia.
