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.
∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx

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Start by rewriting the integral to simplify the integrand. Express the integrand as a sum of separate fractions: \(\int \frac{x + 2\sqrt{x - 1}}{2x\sqrt{x - 1}} \, dx = \int \left( \frac{x}{2x\sqrt{x - 1}} + \frac{2\sqrt{x - 1}}{2x\sqrt{x - 1}} \right) \, dx\).

Simplify each term inside the integral: \(\frac{x}{2x\sqrt{x - 1}} = \frac{1}{2\sqrt{x - 1}}\) and \(\frac{2\sqrt{x - 1}}{2x\sqrt{x - 1}} = \frac{1}{x}\). So the integral becomes \(\int \left( \frac{1}{2\sqrt{x - 1}} + \frac{1}{x} \right) \, dx\).

Split the integral into two separate integrals: \(\int \frac{1}{2\sqrt{x - 1}} \, dx + \int \frac{1}{x} \, dx\).

For the first integral, use the substitution \(u = x - 1\), which implies \(du = dx\). Rewrite the integral as \(\int \frac{1}{2\sqrt{u}} \, du\) and recognize it as a power function integral.

For the second integral, recall that \(\int \frac{1}{x} \, dx = \ln|x| + C\). After integrating both parts, combine the results and add the constant of integration.

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주요 개념

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Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integrand contains composite functions or expressions under roots.

Algebraic Simplification of Integrands

Algebraic simplification involves rewriting the integrand to a more manageable form before integrating. This can include factoring, expanding, or separating terms to isolate simpler expressions. Simplifying the integrand often makes the integral easier to evaluate using standard methods.

Properties of Radicals and Rational Expressions

Understanding how to manipulate radicals and rational expressions is crucial when they appear in integrals. This includes simplifying square roots, rationalizing denominators, and rewriting expressions to facilitate substitution or direct integration. Mastery of these properties helps in transforming complex integrands into integrable forms.

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.∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx