Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.

검증된 단계별 안내

Identify the integral to solve: \(\int \sqrt{1 - x^{2}} \, dx\).

Choose parts for integration by parts: let \(u = \sqrt{1 - x^{2}}\) and \(dv = dx\).

Compute \(du\) and \(v\): differentiate \(u\) to get \(du = \frac{d}{dx} \left( (1 - x^{2})^{1/2} \right) dx = \frac{-x}{\sqrt{1 - x^{2}}} dx\), and integrate \(dv\) to get \(v = x\).

Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), so write \(\int \sqrt{1 - x^{2}} \, dx = x \sqrt{1 - x^{2}} - \int x \left( \frac{-x}{\sqrt{1 - x^{2}}} \right) dx\).

Simplify the integral inside: \(- \int x \left( \frac{-x}{\sqrt{1 - x^{2}}} \right) dx = \int \frac{x^{2}}{\sqrt{1 - x^{2}}} dx\). Then express \(x^{2}\) as \(1 - (1 - x^{2})\) to rewrite the integral and separate it into simpler parts, leading to the formula \(\int \sqrt{1 - x^{2}} \, dx = \frac{1}{2} x \sqrt{1 - x^{2}} + \frac{1}{2} \int \frac{1}{\sqrt{1 - x^{2}}} dx\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

도움이 되었나요?

주요 개념

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

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing appropriate u and dv is crucial to simplify the integral effectively.

Algebraic Manipulation of Integrals

Algebraic manipulation involves rewriting integrals to isolate terms or express them in a more manageable form. In this problem, expressing the integral in terms of itself and another integral helps derive the desired formula, facilitating further evaluation or simplification.

Integrals Involving Square Roots of Quadratic Expressions

Integrals containing expressions like √(1 - x²) often relate to trigonometric substitutions or geometric interpretations. Recognizing these forms helps in choosing substitution methods or integration techniques, as they commonly appear in problems involving circles or arcs.