37–56. Integrals Evaluate each integral.
∫ dx/(8 – x²), x > 2√2

Verified step by step guidance

1. Recognize that the integral is of the form \(\int \frac{dx}{a^2 - x^2}\), where \(a^2 = 8\), so \(a = 2\sqrt{2}\).
2. Recall the standard integral formula: \(\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left| \frac{a + x}{a - x} \right| + C\), valid for \(|x| > a\).
3. Since the problem states \(x > 2\sqrt{2}\), the condition for the formula applies directly.
4. Substitute \(a = 2\sqrt{2}\) into the formula to write the integral in terms of \(x\) and \(a\).
5. Write the final expression for the integral as \(\frac{1}{2 \cdot 2\sqrt{2}} \ln \left| \frac{2\sqrt{2} + x}{2\sqrt{2} - x} \right| + C\), simplifying the coefficient if desired.

Key Concepts

Integration of Rational Functions

This involves integrating functions expressed as ratios of polynomials. Recognizing the form of the integrand helps determine the appropriate method, such as partial fractions or substitution, to simplify and evaluate the integral.

Inverse Hyperbolic Functions

Integrals involving expressions like 1/(a² - x²) often lead to inverse hyperbolic functions such as arctanh or arcsinh. Understanding their definitions and derivatives is essential for correctly evaluating these integrals.

Domain Restrictions and Absolute Values

The condition x > 2√2 restricts the domain, affecting the sign of expressions under square roots or logarithms. Properly handling these restrictions ensures the correct form of the antiderivative and avoids extraneous solutions.