37–56. Integrals Evaluate each integral.
∫ dx/x√(16 + x²)

Verified step by step guidance

1. Recognize that the integral is of the form \(\int \frac{dx}{x \sqrt{16 + x^2}}\), which suggests a trigonometric substitution because of the \(\sqrt{a^2 + x^2}\) term, where \(a = 4\).
2. Use the substitution \(x = 4 \tan(\theta)\), which implies \(dx = 4 \sec^2(\theta) d\theta\). This substitution simplifies the square root because \(\sqrt{16 + x^2} = \sqrt{16 + 16 \tan^2(\theta)} = 4 \sec(\theta)\).
3. Rewrite the integral in terms of \(\theta\) by substituting \(x\) and \(dx\) and simplifying the expression inside the integral.
4. Simplify the resulting integral to a form involving trigonometric functions that can be integrated more easily, such as \(\int \frac{4 \sec^2(\theta) d\theta}{4 \tan(\theta) \cdot 4 \sec(\theta)}\).
5. Integrate the simplified trigonometric integral, then substitute back \(\theta = \tan^{-1}(x/4)\) to express the answer in terms of \(x\).

Key Concepts

Integration involving square roots of quadratic expressions

Integrals containing expressions like √(a² + x²) often require special techniques such as trigonometric substitution to simplify the integrand. Recognizing the form helps in choosing the appropriate substitution to transform the integral into a more manageable form.

Trigonometric substitution

Trigonometric substitution replaces variables with trigonometric functions to simplify integrals involving √(a² + x²), √(a² - x²), or √(x² - a²). For √(a² + x²), substituting x = a tan θ converts the square root into a secant function, facilitating easier integration.

Integration of rational functions after substitution

After substitution, the integral often reduces to a rational function of trigonometric expressions. Understanding how to integrate these resulting functions, such as sec θ or tan θ, is essential to complete the integration and then revert back to the original variable.