(c) Verify that

(a) $x=$ ______ gives $\int dx/\sqrt{x^2-1}=\ln (\sec \theta +\tan \theta )$.

(b) From the triangle, this answer is $f=\ln (x+\sqrt{x^2-1})$. Check that $df/dx=1/\sqrt{x^2-1}$.

(c) Verify that $\cosh f=\frac{1}{2}(e^f+e^{-f})=x$. Then $f={\cosh }^{-1}x$, the answer in Example 8.

This exercise builds on another exercise in its source book, which isn't shown here.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §7.3 Trigonometric Substitutions, #35