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 - √x)

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Start by rewriting the integral to make it easier to work with. Notice that the denominator is \(x - \sqrt{x}\). To simplify, let’s express everything in terms of \(\sqrt{x}\). Set \(t = \sqrt{x}\), which implies \(x = t^2\) and \(dx = 2t \, dt\).

Substitute \(x = t^2\) and \(dx = 2t \, dt\) into the integral. The integral becomes \(\int \frac{2t \, dt}{t^2 - t}\). Simplify the denominator: \(t^2 - t = t(t - 1)\), so the integral is \(\int \frac{2t}{t(t - 1)} \, dt\).

Simplify the integrand by canceling \(t\) in numerator and denominator, resulting in \(\int \frac{2}{t - 1} \, dt\). This is a simpler rational function integral.

Integrate \(\int \frac{2}{t - 1} \, dt\) by recognizing it as a standard form \(\int \frac{1}{u} \, du = \ln|u| + C\). Here, \(u = t - 1\), so the integral is \(2 \ln|t - 1| + C\).

Finally, substitute back \(t = \sqrt{x}\) to express the answer in terms of \(x\). The result is \(2 \ln|\sqrt{x} - 1| + C\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Algebraic Manipulation

Algebraic manipulation involves rewriting expressions to simplify integrals. In this problem, expressing the denominator in terms of a single variable or factoring can make the integral easier to evaluate. For example, substituting √x with a new variable can transform the integral into a more manageable form.

Substitution Method

The substitution method replaces a complicated expression with a simpler variable to facilitate integration. Here, letting t = √x converts the integral into a rational function in terms of t, allowing easier integration. This technique is essential when dealing with roots or composite functions.

Integration of Rational Functions

Integrating rational functions often involves partial fraction decomposition or recognizing standard integral forms. After substitution, the integral becomes a rational function in t, which can be integrated by breaking it into simpler fractions or using known integral formulas.

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 - √x)