The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀⁴ dx / √(4 − x)

검증된 단계별 안내

Identify the integral to be solved: \(\int_0^4 \frac{dx}{\sqrt{4 - x}}\).

Recognize that the integrand involves a square root of a linear function, suggesting a substitution to simplify the integral.

Use the substitution \(u = 4 - x\), which implies \(du = -dx\) or \(dx = -du\). Also, change the limits of integration accordingly: when \(x=0\), \(u=4\); when \(x=4\), \(u=0\).

Rewrite the integral in terms of \(u\): \(\int_{u=4}^{u=0} \frac{-du}{\sqrt{u}} = \int_0^4 \frac{du}{\sqrt{u}}\) after reversing the limits to keep the integral positive.

Evaluate the integral \(\int_0^4 u^{-1/2} du\) by applying the power rule for integration: \(\int u^n du = \frac{u^{n+1}}{n+1} + C\) for \(n \neq -1\).

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주요 개념

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Improper Integrals and Convergence

An integral is improper if the interval of integration is infinite or the integrand becomes unbounded within the interval. Understanding convergence means determining whether the integral approaches a finite value. In this problem, the integrand has a singularity at x = 4, so recognizing the integral as improper is essential.

Substitution Method for Integration

The substitution method simplifies integrals by changing variables to transform the integrand into a more manageable form. For example, setting a new variable equal to the expression inside the square root can help rewrite the integral in terms of a simpler function, making it easier to evaluate.

Evaluating Definite Integrals with Square Root Functions

Integrals involving square roots often require algebraic manipulation or trigonometric substitution to evaluate. Recognizing the form √(a - x) allows the use of substitution or standard integral formulas to find antiderivatives and then apply the limits to compute the definite integral.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₀⁴ dx / √(4 − x)