From my calculus textbook,
I did the substitution u = sec(x) and my final answer was
sec2(x) / 2 + C
but the answer in my textbook was
tan2(x) / 2 + C
This is where the extra constant term becomes crucial. Consider this trig identity:
sec2(x) – 1 = tan2(x)
So the squares of sec(x) and tan(x) differ only by a constant. Which means that the two can be indefinite integrals of the same function (an indefinite integral is a family of functions that differ only by a constant).
Here is a graph of sec2(x) / 2 and tan2(x) / 2. You can think of the white space between the two curves as the constant C that the two indefinite integrals differ by.
This is why thinking in terms of graphs is helpful. It reminds you that math is about ideas, not nomenclature. The symbols “tan(x)” and “sec(x)” don’t give us an intuitive understanding of what’s going on. But a simple graph makes it easy to see how similar sec2(x) and tan2(x) actually are.
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