Evaluate the integrals in Exercises 31–78.
73. ∫dx/√(-2x-x²)

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Rewrite the expression inside the square root to a more recognizable form by completing the square for the quadratic expression: \(-2x - x^2\).

Start by factoring out the negative sign: \(- (x^2 + 2x)\), then complete the square inside the parentheses: \(x^2 + 2x = (x + 1)^2 - 1\).

Substitute back to get the integrand as \(\frac{1}{\sqrt{- ( (x + 1)^2 - 1 )}} = \frac{1}{\sqrt{1 - (x + 1)^2}}\).

Recognize that the integral now has the form \(\int \frac{dx}{\sqrt{a^2 - u^2}}\), which is a standard integral with solution involving inverse trigonometric functions.

Use the substitution \(u = x + 1\) and apply the formula \(\int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin\left(\frac{u}{a}\right) + C\) to express the integral in terms of \(x\).

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주요 개념

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Integration of Functions Involving Square Roots

Integrals containing square roots often require algebraic manipulation or substitution to simplify the integrand. Recognizing the form under the root helps in choosing an appropriate method, such as completing the square or trigonometric substitution, to evaluate the integral effectively.

Completing the Square

Completing the square rewrites a quadratic expression into the form (x + a)² + b, which simplifies the integrand, especially under a square root. This technique transforms the integral into a standard form, making it easier to apply known integration formulas or substitutions.

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots with trigonometric functions, leveraging identities like sin²θ + cos²θ = 1. This method is particularly useful when the integrand contains expressions like √(a² - x²), √(x² - a²), or √(x² + a²), facilitating easier integration.

Evaluate the integrals in Exercises 31–78.73. ∫dx/√(-2x-x²)