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 √(2x - x^2) dx

검증된 단계별 안내

Identify the integral to solve: \(\int x^{2} \sqrt{2x - x^{2}} \, dx\).

Rewrite the expression inside the square root to recognize a substitution: \(2x - x^{2} = -(x^{2} - 2x) = -(x^{2} - 2x + 1 - 1) = -(x - 1)^{2} + 1\).

Use the substitution \(t = x - 1\), so that \(x = t + 1\) and \(dx = dt\). Rewrite the integral in terms of \(t\): \(\int (t + 1)^{2} \sqrt{1 - t^{2}} \, dt\).

Expand \((t + 1)^{2}\) to \(t^{2} + 2t + 1\) and write the integral as \(\int (t^{2} + 2t + 1) \sqrt{1 - t^{2}} \, dt\).

Split the integral into three separate integrals: \(\int t^{2} \sqrt{1 - t^{2}} \, dt + 2 \int t \sqrt{1 - t^{2}} \, dt + \int \sqrt{1 - t^{2}} \, dt\), each of which can be found in standard integral tables or solved using further substitution.

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주요 개념

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

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more familiar form. By choosing an appropriate substitution, such as setting u equal to an expression inside the integral, the integral becomes easier to evaluate using standard techniques or tables.

Integration of Functions Involving Square Roots

Integrals containing square roots often require algebraic manipulation or trigonometric substitution to simplify the integrand. Recognizing patterns like √(a^2 - x^2) or √(2x - x^2) helps in selecting the right substitution to convert the integral into a standard form.

Using Integral Tables

Integral tables list standard integrals and their solutions, providing a quick reference for evaluating common integrals. After substitution simplifies the integral, matching it 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 √(2x - x^2) dx