Evaluate the integrals in Exercises 1–14.
∫ (3 dx) / √(1 + 9x²)

Verified step by step guidance

1. Recognize that the integral has the form \(\int \frac{3 \, dx}{\sqrt{1 + 9x^2}}\), which resembles the standard integral \(\int \frac{dx}{\sqrt{a^2 + x^2}}\) whose antiderivative involves a hyperbolic or inverse hyperbolic function or a logarithm depending on the substitution.
2. Identify the constant inside the square root: here, \(a^2 = 1\) and the term with \(x^2\) is \(9x^2 = (3x)^2\). This suggests using a substitution to simplify the expression under the square root.
3. Make the substitution \(u = 3x\), which implies \(du = 3 \, dx\) or equivalently \(dx = \frac{du}{3}\). This will help rewrite the integral in terms of \(u\).
4. Rewrite the integral in terms of \(u\): substitute \(3 \, dx = du\) and \(\sqrt{1 + 9x^2} = \sqrt{1 + u^2}\), so the integral becomes \(\int \frac{du}{\sqrt{1 + u^2}}\).
5. Recall the antiderivative formula \(\int \frac{du}{\sqrt{1 + u^2}} = \sinh^{-1}(u) + C\) or equivalently \(\ln|u + \sqrt{1 + u^2}| + C\). After integrating, substitute back \(u = 3x\) to express the answer in terms of \(x\).

Key Concepts

Integration of Functions Involving Square Roots

This concept involves integrating functions where the integrand contains a square root expression, often requiring substitution or recognition of standard integral forms. Understanding how to manipulate and simplify the integrand is crucial for solving such integrals.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² + x²), √(a² - x²), or √(x² - a²). By substituting x with a trigonometric function, the integral becomes easier to evaluate using trigonometric identities.

Standard Integral Formulas

Familiarity with standard integral formulas, such as ∫ dx / √(a² + x²) = ln|x + √(x² + a²)| + C, helps quickly evaluate integrals without lengthy calculations. Recognizing these forms allows for efficient problem-solving.