7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
26. ∫[√2 to √2] √(x² - 1)/x 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 a trigonometric substitution. Specifically, let x = sec(θ), where sec²(θ) - 1 = tan²(θ). This substitution simplifies the square root.
2. Step 2: Substitute x = sec(θ) into the integral. Compute dx = sec(θ)tan(θ)dθ and replace √(x² - 1) with √(sec²(θ) - 1) = tan(θ). The integral becomes ∫ √(x² - 1)/x dx = ∫ tan(θ)/sec(θ) * sec(θ)tan(θ)dθ.
3. Step 3: Simplify the integrand. The expression tan(θ)/sec(θ) * sec(θ)tan(θ) simplifies to tan²(θ)dθ. The integral now becomes ∫ tan²(θ)dθ.
4. Step 4: Use the trigonometric identity tan²(θ) = sec²(θ) - 1 to rewrite the integral. The integral becomes ∫ (sec²(θ) - 1)dθ, which can be split into two separate integrals: ∫ sec²(θ)dθ - ∫ 1dθ.
5. Step 5: Evaluate the two integrals separately. ∫ sec²(θ)dθ = tan(θ), and ∫ 1dθ = θ. After integration, convert back to the original variable x using the substitution x = sec(θ), and apply the given limits of integration √2 to √2.

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 x = sec(θ) or x = sin(θ), the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(x² - a²), √(a² - x²), or √(x² + a²).

Integration Techniques

Integration techniques encompass various methods used to evaluate integrals, including substitution, integration by parts, and partial fractions. Understanding these techniques is essential for solving complex integrals, as they provide strategies to break down the integral into simpler parts. Mastery of these methods allows students to tackle a wide range of integrals, including those that arise from trigonometric substitutions.

Definite Integrals

Definite integrals represent the accumulation of quantities, calculated over a specific interval [a, b]. They are denoted as ∫[a to b] f(x) dx and yield a numerical value that corresponds to the area under the curve of the function f(x) between the limits a and b. Evaluating definite integrals often involves finding the antiderivative of the function and applying the Fundamental Theorem of Calculus, which connects differentiation and integration.