In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 / √(x^2 - 4x + 5) dx

검증된 단계별 안내

Start by examining the expression inside the square root: \(x^2 - 4x + 5\). To simplify this, complete the square for the quadratic expression.

Rewrite \(x^2 - 4x + 5\) as \((x^2 - 4x + 4) + 1\), which simplifies to \((x - 2)^2 + 1\).

Use the substitution \(u = x - 2\), which implies \(du = dx\). This changes the integral to an expression in terms of \(u\).

Rewrite the integral in terms of \(u\): replace \(x^2\) with \((u + 2)^2\) and \(\sqrt{x^2 - 4x + 5}\) with \(\sqrt{u^2 + 1}\). The integral becomes \(\int \frac{(u + 2)^2}{\sqrt{u^2 + 1}} \, du\).

Expand \((u + 2)^2\) to \(u^2 + 4u + 4\) and split the integral into three separate integrals: \(\int \frac{u^2}{\sqrt{u^2 + 1}} \, du + 4 \int \frac{u}{\sqrt{u^2 + 1}} \, du + 4 \int \frac{1}{\sqrt{u^2 + 1}} \, du\). Each of these can be found in standard integral tables.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

13m

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 the integrand into a form that is easier to integrate. By choosing an appropriate substitution, often based on the inner function or expression, the integral can be transformed into a standard form found in integral tables.

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in the form ax^2 + bx + c as (x - h)^2 + k. This form simplifies the integrand, especially under square roots, making it easier to identify substitutions or match standard integral forms.

Using Integral Tables

Integral tables list standard integrals and their antiderivatives, providing quick references for common integral forms. After substitution and simplification, matching the integral to a form in the table allows for direct evaluation without performing integration from first principles.

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.∫ x^2 / √(x^2 - 4x + 5) dx