Evaluate the integrals in Exercises 53–76.
69. ∫dx/((2x-1)√((2x-1)²-4))

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Identify the integral to solve: \(\int \frac{dx}{(2x-1) \sqrt{(2x-1)^2 - 4}}\).

Make a substitution to simplify the integral. Let \(u = 2x - 1\), so that \(du = 2 \, dx\) or \(dx = \frac{du}{2}\).

Rewrite the integral in terms of \(u\): \(\int \frac{\frac{du}{2}}{u \sqrt{u^2 - 4}} = \frac{1}{2} \int \frac{du}{u \sqrt{u^2 - 4}}\).

Recognize the integral form \(\int \frac{du}{u \sqrt{u^2 - a^2}}\), which suggests using a trigonometric substitution such as \(u = 2 \sec \theta\) to simplify the square root.

Perform the substitution \(u = 2 \sec \theta\), find \(du\), and rewrite the integral in terms of \(\theta\) to proceed with integration.

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주요 개념

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Integration involving square roots of quadratic expressions

Integrals containing expressions like √(ax + b)² - c² often require recognizing the form as a difference of squares under the root. This suggests using trigonometric or hyperbolic substitutions to simplify the integral by transforming the radical into a simpler expression.

Trigonometric substitution

Trigonometric substitution replaces variables with trigonometric functions to simplify integrals involving radicals such as √(x² - a²). For example, setting x = a sec θ transforms the radical into a trigonometric expression, making the integral easier to evaluate.

Substitution method (u-substitution)

U-substitution involves changing variables to simplify the integral, especially when the integrand contains a composite function. Identifying a part of the integrand as u and expressing dx in terms of du can reduce the integral to a basic form that is easier to integrate.

Evaluate the integrals in Exercises 53–76.69. ∫dx/((2x-1)√((2x-1)²-4))