7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
36. ∫[8√2 to 16] 1/√(x² - 64) dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves a square root of the form √(x² - a²), which suggests using the trigonometric substitution x = a sec(θ). Here, a = 8 because √(x² - 64) can be rewritten as √(x² - 8²). Substitute x = 8 sec(θ).
2. Step 2: Compute dx using the substitution x = 8 sec(θ). The derivative of x = 8 sec(θ) is dx = 8 sec(θ) tan(θ) dθ. Replace dx in the integral with this expression.
3. Step 3: Substitute x = 8 sec(θ) into √(x² - 64). Using the identity sec²(θ) - 1 = tan²(θ), √(x² - 64) becomes √(64 sec²(θ) - 64) = √(64 tan²(θ)) = 8 tan(θ). Replace √(x² - 64) in the integral with 8 tan(θ).
4. Step 4: Update the limits of integration. When x = 8√2, solve for θ using x = 8 sec(θ): sec(θ) = √2, so θ = π/4. When x = 16, solve for θ: sec(θ) = 2, so θ = π/3. The new limits of integration are θ = π/4 to θ = π/3.
5. Step 5: Simplify the integral. After substitution, the integral becomes ∫[π/4 to π/3] (1 / (8 tan(θ))) * (8 sec(θ) tan(θ)) dθ. Simplify the expression to ∫[π/4 to π/3] sec(θ) dθ. Evaluate this integral using the antiderivative of sec(θ), which is ln|sec(θ) + tan(θ)|, and apply the limits of integration.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as sine or cosine, the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(a² - x²), √(x² - a²), or √(x² + a²).

Integral of a Function

The integral of a function represents the area under the curve of that function over a specified interval. In this context, evaluating the integral ∫[8√2 to 16] 1/√(x² - 64) dx involves finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus to compute the definite integral. Understanding how to evaluate integrals is crucial for solving problems in calculus.

Definite Integral

A definite integral is an integral that computes the accumulation of quantities, such as area, over a specific interval [a, b]. It is represented as ∫[a to b] f(x) dx, where f(x) is the integrand. The result of a definite integral is a numerical value that reflects the total area between the curve and the x-axis from x = a to x = b. This concept is essential for applying limits and evaluating integrals in practical scenarios.