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

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Start by rewriting the expression under the square root to a more recognizable form. The integrand is \(\frac{1}{\sqrt{-x^{2} + 4x - 3}}\). Notice the quadratic inside the square root: \(-x^{2} + 4x - 3\).

Complete the square for the quadratic expression inside the square root. Factor out the negative sign first: \(-(x^{2} - 4x + 3)\). Then complete the square for \(x^{2} - 4x + 3\) by finding a perfect square trinomial.

Recall that \(x^{2} - 4x + 3 = (x - 2)^{2} - 1\) because \((x - 2)^{2} = x^{2} - 4x + 4\). Substitute this back to get \(-( (x - 2)^{2} - 1 ) = 1 - (x - 2)^{2}\) inside the square root.

Rewrite the integral as \(\int \frac{dx}{\sqrt{1 - (x - 2)^{2}}}\). This is a standard form that resembles the integral of \(\frac{1}{\sqrt{a^{2} - u^{2}}}\), which is related to the inverse sine function.

Use the substitution \(u = x - 2\), so \(du = dx\). The integral becomes \(\int \frac{du}{\sqrt{1 - u^{2}}}\). Recognize this as the integral of \(\frac{1}{\sqrt{1 - u^{2}}}\), whose antiderivative is \(\arcsin(u) + C\). Finally, substitute back \(u = x - 2\) to express the answer in terms of \(x\).

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주요 개념

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Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in the form ax² + bx + c as (x - h)² + k. This simplifies the integrand, especially when dealing with square roots of quadratic expressions, making it easier to identify the integral's form and apply appropriate methods.

Integration of Functions Involving Square Roots

Integrals containing square roots of quadratic expressions often require substitution or recognition of standard integral forms. Understanding how to manipulate the integrand into a recognizable pattern, such as arcsine or logarithmic forms, is essential for evaluating these integrals.

Trigonometric Substitution

Trigonometric substitution replaces variables in integrals involving square roots of quadratic expressions with trigonometric functions, simplifying the integral. For example, substituting x = a sin θ or x = a cos θ transforms the square root into a trigonometric identity, facilitating easier integration.

Evaluate the integrals in Exercises 77–90.77. ∫dx/√(-x²+4x-3)