60–69. Completing the square Evaluate the following integrals.
68. ∫ dx / sqrt((x - 1)(3 - x))

검증된 단계별 안내

Step 1: Recognize that the integral involves a square root in the denominator, specifically sqrt((x - 1)(3 - x)). This suggests that completing the square or trigonometric substitution might be useful.

Step 2: Rewrite the expression (x - 1)(3 - x) in a more convenient form. Factor it as -(x^2 - 4x + 3). Then complete the square for the quadratic expression inside the parentheses: x^2 - 4x + 3 = (x - 2)^2 - 1.

Step 3: Substitute the completed square form back into the integral. The denominator becomes sqrt(-((x - 2)^2 - 1)), which simplifies to sqrt(1 - (x - 2)^2). This suggests a trigonometric substitution.

Step 4: Use the substitution x - 2 = sin(θ), which implies dx = cos(θ)dθ and simplifies the square root to sqrt(1 - sin^2(θ)) = cos(θ). Substitute these into the integral.

Step 5: After substitution, the integral becomes ∫ dθ, which is straightforward to evaluate. Once the integral is solved, back-substitute to return to the original variable x.

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Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is essential for simplifying integrals involving square roots of quadratic expressions, allowing for easier integration. By rewriting the expression in the form (x - h)² - k, we can identify the vertex and facilitate the integration process.

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots. By substituting a variable with a trigonometric function, such as x = a sin(θ) or x = a cos(θ), we can transform the integral into a more manageable form. This method is particularly useful for integrals that involve expressions like √(a² - x²) or √(x² - a²).

Definite and Indefinite Integrals

Integrals can be classified as definite or indefinite. An indefinite integral represents a family of functions and includes a constant of integration, while a definite integral calculates the area under a curve between two specific limits. Understanding the distinction is crucial for evaluating integrals correctly, especially when applying techniques like substitution or completing the square.

60–69. Completing the square Evaluate the following integrals.68. ∫ dx / sqrt((x - 1)(3 - x))