The Joy of ∫sec x dx

Circular and hyperbolic trigonometry; a logarithm and complex exponential.

Study the mathematical smorgasbord below. Do you know how these expressions are all connected?

Let’s see.

We’ll begin with the antiderivative.

Ready?

A. Without Exponentials

1. Who thinks in secants? Let’s start with the reciprocal.

2. Maybe log something? We need a derivative in the numerator.

3. Now substitute:

4. Where to now? Choose your own adventure — a or b?

a) Partial Fractions

i. The denominator is a second-degree polynomial. As long as the numerator looks like a first-degree polynomial, we’re okay. A coefficient of 0 takes care of that.