The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dx / ((x - 2)√(x² - 4x + 3)))

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Start by simplifying the expression under the square root. Notice that \(x^2 - 4x + 3\) can be factored or completed the square. Rewrite it as \(x^2 - 4x + 3 = (x - 2)^2 - 1\).

Make the substitution \(u = x - 2\) to simplify the integral. Then, \(du = dx\), and the integral becomes \(\int \frac{du}{u \sqrt{u^2 - 1}}\).

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

Express \(du\) in terms of \(d\theta\): since \(u = \sec \theta\), then \(du = \sec \theta \tan \theta \, d\theta\). Substitute these into the integral to rewrite it entirely in terms of \(\theta\).

Simplify the resulting integral in \(\theta\) and integrate using standard trigonometric integral formulas. 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 letting a new variable represent a part of the integrand, the integral can often be transformed into a more manageable form. This technique is especially useful when the integrand contains composite functions or expressions under radicals.

Simplifying Radicals and Quadratic Expressions

Simplifying the expression under the square root, often a quadratic, helps in recognizing patterns or suitable substitutions. Completing the square or factoring the quadratic can transform the radical into a standard form, making the integral easier to evaluate using known formulas or trigonometric substitutions.

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots of quadratic polynomials with trigonometric functions. This method leverages identities like sin²θ + cos²θ = 1 to simplify integrals containing radicals, converting them into integrals of trigonometric functions that are easier to solve.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.∫ (dx / ((x - 2)√(x² - 4x + 3)))