In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ (√2 - x) / √x dx

검증된 단계별 안내

Identify a substitution that simplifies the integral. Since the integral contains both \( \sqrt{x} \) and \( x \), let’s set \( u = \sqrt{x} \). This means \( u = x^{1/2} \).

Express \( x \) and \( dx \) in terms of \( u \). Since \( u = x^{1/2} \), then \( x = u^2 \). Differentiate both sides with respect to \( u \) to find \( dx \): \( dx = 2u \, du \).

Rewrite the integral in terms of \( u \). Substitute \( \sqrt{x} = u \), \( x = u^2 \), and \( dx = 2u \, du \) into the integral: \[ \int \frac{\sqrt{2} - x}{\sqrt{x}} \, dx = \int \frac{\sqrt{2} - u^2}{u} \cdot 2u \, du \].

Simplify the integrand by canceling and combining terms: \( \frac{\sqrt{2} - u^2}{u} \cdot 2u = 2(\sqrt{2} - u^2) \). So the integral becomes \( \int 2(\sqrt{2} - u^2) \, du \).

Now, split the integral into two simpler integrals: \[ 2 \int \sqrt{2} \, du - 2 \int u^2 \, du \]. These are standard integrals that can be found in the integral table and evaluated easily.

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.

Substitution Method in Integration

The substitution method involves changing variables in an integral to simplify it into a more familiar form. By letting a new variable equal a function of the original variable, the integral can be rewritten and evaluated more easily, often matching standard integral forms.

Manipulating Radicals and Algebraic Expressions

Understanding how to rewrite expressions involving square roots and algebraic terms is essential. This includes expressing radicals as fractional exponents and simplifying the integrand to facilitate substitution and integration.

Using Integral Tables

Integral tables provide formulas for common integrals, allowing quick evaluation once the integral is transformed appropriately. Recognizing the form of the integral after substitution helps in matching it to a table entry for direct integration.

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.∫ (√2 - x) / √x dx