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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.6.90
Chapter 7, Problem 7.6.90

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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Substitution With an Extra Variable

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.
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Completing the Square to Rewrite the Integrand

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.
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Integrals Involving Natural Logs: Substitution Example 7
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