The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|

검증된 단계별 안내

First, recognize that the integral involves the expression \(\sqrt{|x - 1|}\), which has an absolute value inside the square root. This means the behavior of the integrand changes at \(x = 1\), so we should split the integral at this point.

Rewrite the integral as the sum of two integrals: \(\int_0^1 \frac{dx}{\sqrt{1 - x}} + \int_1^2 \frac{dx}{\sqrt{x - 1}}\). Notice how the absolute value affects the expression inside the square root on each interval.

For the first integral, \(\int_0^1 \frac{dx}{\sqrt{1 - x}}\), use the substitution \(u = 1 - x\), which implies \(du = -dx\). Adjust the limits accordingly and rewrite the integral in terms of \(u\).

For the second integral, \(\int_1^2 \frac{dx}{\sqrt{x - 1}}\), use the substitution \(v = x - 1\), which implies \(dv = dx\). Adjust the limits accordingly and rewrite the integral in terms of \(v\).

Evaluate both integrals using the power rule for integrals: \(\int u^{n} du = \frac{u^{n+1}}{n+1} + C\) for \(n \neq -1\). After evaluating, sum the results to get the value of the original integral.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Improper Integrals

Improper integrals involve integrands with infinite discontinuities or infinite limits of integration. In this problem, the integrand has a singularity at x = 1, where the denominator becomes zero, requiring careful evaluation of the integral as a limit approaching the point of discontinuity.

Absolute Value and Piecewise Functions

The absolute value inside the square root splits the integral into separate intervals where the expression inside changes sign. Understanding how to rewrite |x - 1| as (1 - x) or (x - 1) depending on the interval is essential to correctly set up and evaluate the integral.

Integration Techniques for Power Functions

Integrating functions of the form 1/√(x - a) involves recognizing them as power functions with fractional exponents. Applying the power rule for integration, ∫x^n dx = x^(n+1)/(n+1), is necessary after rewriting the integrand appropriately to find the antiderivative.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₀² dx / √|x − 1|