Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 19

$$\frac{1}{5} \sinh ^{-1}\left(\frac{5 x}{4}\right)+C $$

\begin{aligned} \int \frac{1}{\sqrt{16+25 x^{2}}} d x &=\int \frac{1}{\sqrt{16\left(1+\frac{25}{16} x^{2}\right)}} d x \\ &=\int \frac{1}{4 \sqrt{1+\left(\frac{5 x}{4}\right)^{2}}} d x \\ &=\frac{A}{5} \int \frac{\frac{5}{4}}{\sqrt{1+\left(\frac{5 x}{4}\right)^{2}}} d x \\ &=\frac{1}{5} \int \frac{\frac{5}{4}}{\sqrt{1+\left(\frac{5 x}{4}\right)^{2}}} d x \\ &=\frac{1}{5} \sinh ^{-1}\left(\frac{5 x}{4}\right)+C \end{aligned}