2–9. Integrals Evaluate the following integrals.


∫ dx / √(x² − 9),x > 3

Verified step by step guidance

1. Recognize that the integral is of the form \(\int \frac{dx}{\sqrt{x^2 - a^2}}\) where \(a = 3\). This is a standard integral involving a square root of a difference of squares.
2. Recall the standard formula for this integral: \(\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln|x + \sqrt{x^2 - a^2}| + C\), where \(C\) is the constant of integration.
3. Since the problem states \(x > 3\), the expression inside the logarithm is positive, so the absolute value can be dropped for this domain.
4. Write the integral solution using the formula with \(a = 3\): \(\int \frac{dx}{\sqrt{x^2 - 9}} = \ln(x + \sqrt{x^2 - 9}) + C\).
5. Verify the solution by differentiating \(\ln(x + \sqrt{x^2 - 9})\) to ensure it matches the original integrand \(\frac{1}{\sqrt{x^2 - 9}}\).

Key Concepts

Integration of Functions Involving Square Roots

This concept involves techniques for integrating functions that contain square roots, especially expressions like √(x² − a²). Recognizing the form helps in choosing appropriate substitution methods or formulas to simplify the integral.

Inverse Hyperbolic Functions

Integrals of the form ∫ dx / √(x² − a²) often result in inverse hyperbolic functions such as arcosh(x/a). Understanding these functions and their derivatives is essential for evaluating and expressing the integral's solution.

Domain Restrictions and Absolute Values

The condition x > 3 ensures the expression under the square root is positive, affecting the integral's domain and the form of the solution. Recognizing domain restrictions helps avoid extraneous solutions and correctly apply absolute values in logarithmic or inverse hyperbolic expressions.