# Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 29

$$\cosh ^{-1} x-\frac{\sqrt{x^{2}-1}}{x}+C$$

#### Work Step by Step

Given $$\int \frac{\sqrt{x^{2}-1}}{x^{2}} d x$$ Let $$x=\cosh t\ \ \to \ dx=\sinh t dt$$ \begin{aligned} \int \frac{\sqrt{x^{2}-1}}{x^{2}} d x &=\int \frac{\sqrt{\cosh ^{2} t-1}}{\cosh ^{2} t} \cdot \sinh t d t \\ &=\int \frac{\sqrt{\sinh ^{2} t}}{\cosh ^{2} t} \cdot \sinh t d t \\ &=\int \frac{\sinh ^{2} t}{\cosh ^{2} t} d t \\ &=\int \tanh ^{2} t d t \\ &=\int\left(1-\operatorname{sech}^{2} t\right) d t \\ &=t-\tanh t +C\\ &= \cosh ^{-1} x-\frac{\sinh t}{\cosh t}+C\\ &=\cosh ^{-1} x-\frac{\sqrt{x^{2}-1}}{x}+C \end{aligned}

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