Evaluate the integrals in Exercises 77–90.
90. ∫dx/((x-2)√(x²-4x+3))

검증된 단계별 안내

Start by simplifying the expression inside the square root: rewrite \(x^2 - 4x + 3\) in a completed square form. To do this, complete the square for the quadratic: \(x^2 - 4x + 3 = (x - 2)^2 - 1\).

Substitute \(u = x - 2\) to simplify the integral. This changes the integral to \(\int \frac{dx}{(x-2) \sqrt{(x-2)^2 - 1}} = \int \frac{du}{u \sqrt{u^2 - 1}}\) since \(dx = du\).

Recognize that the integral now has the form \(\int \frac{du}{u \sqrt{u^2 - 1}}\), which suggests using a trigonometric substitution. Use \(u = \sec \theta\), so that \(\sqrt{u^2 - 1} = \tan \theta\) and \(du = \sec \theta \tan \theta \, d\theta\).

Rewrite the integral in terms of \(\theta\): substitute \(u\), \(du\), and \(\sqrt{u^2 - 1}\) into the integral to get \(\int \frac{\sec \theta \tan \theta \, d\theta}{\sec \theta \cdot \tan \theta}\), which simplifies the integrand.

Simplify the integral and integrate with respect to \(\theta\). After integration, substitute back \(\theta = \sec^{-1}(u)\) and then \(u = x - 2\) to express the answer in terms of \(x\).

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주요 개념

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Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By identifying a part of the integrand as a new variable, the integral can be transformed into a more manageable form. This technique is especially useful when the integrand contains composite functions or expressions inside roots.

Completing the Square

Completing the square rewrites a quadratic expression into the form (x - h)² + k, making it easier to analyze or integrate. This method helps simplify expressions under square roots, allowing the use of standard integral formulas involving square roots of perfect squares or shifted variables.

Integrals Involving Square Roots of Quadratic Expressions

Integrals containing square roots of quadratic expressions often require special techniques such as trigonometric substitution or recognizing standard integral forms. Understanding how to handle √(ax² + bx + c) is crucial, as it allows the integral to be expressed in terms of inverse trigonometric or logarithmic functions.

Evaluate the integrals in Exercises 77–90.90. ∫dx/((x-2)√(x²-4x+3))