1. The problem is to find an integral whose result is the inverse secant function, $\arcsec(x)$. 2. Recall that the derivative of $\arcsec(x)$ is given by the formula: $$\frac{d}{dx} \arcsec(x) = \frac{1}{|x| \sqrt{x^2 - 1}}$$ 3. Therefore, the integral we want is: $$\int \frac{1}{|x| \sqrt{x^2 - 1}} \, dx = \arcsec(x) + C$$ 4. This means the function inside the integral is the derivative of $\arcsec(x)$, so integrating it returns $\arcsec(x)$ plus a constant of integration. 5. In summary, the integral that makes $\arcsec(x)$ is: $$\int \frac{1}{|x| \sqrt{x^2 - 1}} \, dx = \arcsec(x) + C$$