These answers agree

(a) If $x=\tan \theta$ then $\int \sqrt{1+x^2}dx=\int$ ____ $d\theta$.
(b) Convert $\frac{1}{2}[\sec \theta \tan \theta +\ln (\sec \theta +\tan \theta )]$ back to $x$.
(c) If $x=\sinh \theta$ then $\int \sqrt{1+x^2}dx=\int$ ____ $d\theta$.
(d) Convert $\frac{1}{2}[\sinh \theta \cosh \theta +\theta ]$ back to $x$.

These answers agree. In Section 8.2 they will give the length of a parabola. Compare with Problem 7.2.62.

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, #54