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 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.

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