Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (x² - 1)^(5/2), where x > 1

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Identify the integral to solve: \(\displaystyle \int \frac{x^{2}}{(x^{2} - 1)^{5/2}} \, dx\), with the condition \(x > 1\).

Recognize that the integrand contains the expression \(x^{2} - 1\) under a fractional power, suggesting a trigonometric substitution related to \(\sec \theta\), since \(\sec^{2} \theta - 1 = \tan^{2} \theta\).

Make the substitution \(x = \sec \theta\), which implies \(dx = \sec \theta \tan \theta \, d\theta\). Also, rewrite the denominator: \(x^{2} - 1 = \sec^{2} \theta - 1 = \tan^{2} \theta\).

Rewrite the integral entirely in terms of \(\theta\): replace \(x^{2}\) with \(\sec^{2} \theta\), \((x^{2} - 1)^{5/2}\) with \((\tan^{2} \theta)^{5/2} = \tan^{5} \theta\), and \(dx\) with \(\sec \theta \tan \theta \, d\theta\). Simplify the resulting expression before integrating.

After simplification, integrate the resulting trigonometric expression with respect to \(\theta\). Once integrated, substitute back \(\theta = \sec^{-1} x\) to express the answer in terms of \(x\).

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주요 개념

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Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(x² - a²), √(a² - x²), or √(x² + a²). By substituting x with a trigonometric function, the integral transforms into a trigonometric integral that is easier to evaluate. For example, for √(x² - 1), substituting x = sec(θ) is common.

Integration of Rational Functions

Integrals involving rational functions, where the integrand is a ratio of polynomials, often require algebraic manipulation such as polynomial division or rewriting the integrand to simplify the expression before integrating. Recognizing when to apply these techniques helps in breaking down complex integrals.

Differential and Substitution Method

The substitution method involves changing variables to simplify the integral. After choosing an appropriate substitution (like x = sec(θ)), the differential dx is expressed in terms of dθ, allowing the integral to be rewritten in a simpler form. Correctly handling the differential is crucial for accurate integration.

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.∫ (x² dx) / (x² - 1)^(5/2), where x > 1