Evaluate ∫ x³ √(1 - x²) dx using:
a. Integration by parts.

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Identify the integral to solve: \(\int x^{3} \sqrt{1 - x^{2}} \, dx\).

For integration by parts, choose parts of the integrand as \(u\) and \(dv\). A good choice is to let \(u = x^{2}\) (since \(x^{3} = x \cdot x^{2}\)) and \(dv = x \sqrt{1 - x^{2}} \, dx\) to simplify the integral after differentiation and integration.

Compute \(du\) by differentiating \(u\): \(du = 2x \, dx\).

Find \(v\) by integrating \(dv\): \(v = \int x \sqrt{1 - x^{2}} \, dx\). To do this, use a substitution such as \(t = 1 - x^{2}\), then express \(v\) in terms of \(t\) and integrate.

Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Substitute the expressions for \(u\), \(v\), and \(du\) into this formula to rewrite the original integral in terms of simpler integrals to evaluate.

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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 u and dv wisely simplifies the integral, especially when one function becomes simpler upon differentiation.

Substitution Method

Substitution involves changing variables to simplify an integral. By setting a part of the integrand as a new variable, the integral can be rewritten in a simpler form. This is particularly useful when the integrand contains composite functions, such as √(1 - x²), which suggests substituting u = 1 - x².

Handling Powers and Roots in Integrals

Integrals involving powers and roots require careful algebraic manipulation. Expressing roots as fractional exponents and simplifying powers can make integration more straightforward. Recognizing how to rewrite expressions like √(1 - x²) as (1 - x²)^(1/2) helps in applying integration techniques effectively.

Evaluate ∫ x³ √(1 - x²) dx using:a. Integration by parts.