The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / [x√(x² − 1)]

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Identify the integral to be evaluated: \(\displaystyle \int_1^{\infty} \frac{dx}{x \sqrt{x^2 - 1}}\).

Recognize that the integrand involves \(\sqrt{x^2 - 1}\), which suggests a trigonometric substitution such as \(x = \sec(\theta)\), because \(\sec^2(\theta) - 1 = \tan^2(\theta)\).

Perform the substitution \(x = \sec(\theta)\), then compute \(dx = \sec(\theta) \tan(\theta) d\theta\). Also, rewrite the integrand in terms of \(\theta\):

\[\frac{1}{x \sqrt{x^2 - 1}} dx = \frac{1}{\sec(\theta) \sqrt{\sec^2(\theta) - 1}} \cdot \sec(\theta) \tan(\theta) d\theta.\]

Simplify the expression inside the integral using the identity \(\sqrt{\sec^2(\theta) - 1} = \tan(\theta)\), and then simplify the integrand to a function of \(\theta\) that is easier to integrate.

Change the limits of integration from \(x\) to \(\theta\) using \(x = \sec(\theta)\), then integrate with respect to \(\theta\). Finally, substitute back to \(x\) to express the answer in terms of the original variable.

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주요 개념

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Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, we replace the infinite limit with a variable, compute the integral, and then take the limit as the variable approaches infinity to determine convergence and value.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integrand into a more manageable form. Choosing an appropriate substitution, such as a trigonometric or hyperbolic function, can help evaluate integrals involving expressions like √(x² − 1).

Trigonometric Identities and Inverse Functions

Trigonometric identities relate expressions involving squares and roots, such as x² − 1, to trigonometric functions like secant and tangent. Recognizing these allows rewriting the integral in terms of inverse trigonometric functions, facilitating evaluation without tables.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₁^∞ dx / [x√(x² − 1)]