Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x √(x + 4))

검증된 단계별 안내

Identify the integral to solve: \(\int \frac{dx}{x \sqrt{x + 4}}\).

Consider a substitution to simplify the square root expression. Let \(t = \sqrt{x + 4}\), which implies \(t^2 = x + 4\).

Differentiate both sides with respect to \(x\) to find \(dx\) in terms of \(dt\): \(2t \frac{dt}{dx} = 1\), so \(dx = 2t \, dt\).

Rewrite the integral in terms of \(t\): replace \(x\) with \(t^2 - 4\) and \(dx\) with \(2t \, dt\), then simplify the integrand accordingly.

Use the table of integrals to find the integral of the resulting expression in \(t\), then substitute back \(t = \sqrt{x + 4}\) to express the answer in terms of \(x\).

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주요 개념

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Integration Using Standard Integral Tables

Integral tables provide formulas for common integrals, allowing quick evaluation without performing integration from first principles. Recognizing the integral's form helps match it to a known formula, simplifying the process.

Substitution Method in Integration

Substitution involves changing variables to simplify an integral into a standard form. For integrals with expressions like √(x + a), substituting u = x + a often transforms the integral into a more manageable expression.

Integrals Involving Radical Expressions

Integrals containing radicals such as √(x + c) often require special techniques or formulas. Understanding how to handle these radicals, including rationalizing or using trigonometric substitutions, is essential for evaluating such integrals.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ dx / (x √(x + 4))