Solve each of the integrals . Some integrals require...

Solve each of the integrals . Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
$$
\int \frac{\sqrt{x^{2}-1}}{x} d x
$$

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Key Concepts

u-Substitution

This technique involves choosing a new variable to simplify the integrand by identifying a function within the integrand whose derivative is also present. In the context of integrals like the one given, recognizing that part of the integrand can be represented as the derivative of another function (for instance, setting u = x² - 1) can transform the integral into a simpler form that is easier to evaluate.

Trigonometric Substitution

Trigonometric substitution is a method used to simplify integrals containing square roots of quadratic expressions. When the integrand involves a term like ?(x² - 1), substituting x with a trigonometric function (such as x = sec?) converts the expression under the square root into a trigonometric identity, thereby simplifying the integration process and allowing the use of standard trigonometric integrals.

Back-Substitution

Once the integral has been solved in terms of the substitution variable (whether from u-substitution or trigonometric substitution), back-substitution is the process of converting the result back to the original variable. This step ensures that the final answer is expressed in terms of the original variable of the problem and, where possible, in a purely algebraic form.

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