Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (x² √(x² + 1))

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Identify the integral to solve: \(\int \frac{dx}{x^{2} \sqrt{x^{2} + 1}}\).

Recognize that the integrand contains \(\sqrt{x^{2} + 1}\), which suggests using a trigonometric substitution where \(x = \tan(\theta)\) because \(1 + \tan^{2}(\theta) = \sec^{2}(\theta)\).

Substitute \(x = \tan(\theta)\), then compute \(dx = \sec^{2}(\theta) d\theta\). Also, rewrite the integral in terms of \(\theta\): replace \(x^{2}\) with \(\tan^{2}(\theta)\) and \(\sqrt{x^{2} + 1}\) with \(\sqrt{\tan^{2}(\theta) + 1} = \sec(\theta)\).

Rewrite the integral as \(\int \frac{\sec^{2}(\theta) d\theta}{\tan^{2}(\theta) \cdot \sec(\theta)} = \int \frac{\sec^{2}(\theta)}{\tan^{2}(\theta) \sec(\theta)} d\theta = \int \frac{\sec(\theta)}{\tan^{2}(\theta)} d\theta\).

Simplify the integrand and use trigonometric identities to express everything in terms of sine and cosine, then integrate with respect to \(\theta\). After integration, substitute back \(\theta = \arctan(x)\) to express the answer in terms of \(x\).

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Trigonometric Substitution

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

Integration of Rational Functions

Integrals involving rational functions, especially those with polynomial expressions in the numerator and denominator, often require algebraic manipulation or substitution. Recognizing the structure of the integrand helps in choosing the right substitution or method to simplify the integral.

Fundamental Integration Techniques

Basic integration methods such as substitution, integration by parts, and recognizing standard integral forms are essential. These techniques provide the foundation for solving more complex integrals, including those involving trigonometric substitutions or rational functions.

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.∫ dx / (x² √(x² + 1))