We now turn to a little-taught and perhaps controversial inverse function: arcsecant. As we’ve seen throughout this series, the domain of this inverse function must be chosen so that the graph of satisfies the horizontal line test. It turns out that the choice of domain has surprising consequences that are almost unforeseeable using only the tools of Precalculus.

The standard definition of uses the interval , so that

Why would this be controversial? Yesterday, we saw that is both positive and negative on the interval , and so great care has to be used to calculate the integral:

Here’s another example: let’s use trigonometric substitution to calculate

The standard trick is to use the substitution . With this substitution:

Therefore,

At this point, the common mistake would be to replace with . This is a mistake because

Furthermore, for this particular problem, is negative on the interval . Therefore, for this problem, we should replace with .

Continuing the calculation,

So if great care wasn’t used to correctly simplify , one would instead obtain the incorrect answer .

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