Evaluate ∫ x³ √(1 - x²) dx using:
c. A trigonometric substitution.

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Identify the integral to solve: \(\int x^{3} \sqrt{1 - x^{2}} \, dx\) and recognize that the expression under the square root, \(1 - x^{2}\), suggests a trigonometric substitution involving sine or cosine.

Use the substitution \(x = \sin(\theta)\), which implies \(dx = \cos(\theta) \, d\theta\). This substitution transforms the square root as \(\sqrt{1 - x^{2}} = \sqrt{1 - \sin^{2}(\theta)} = \cos(\theta)\).

Rewrite the integral in terms of \(\theta\): replace \(x^{3}\) with \(\sin^{3}(\theta)\), \(\sqrt{1 - x^{2}}\) with \(\cos(\theta)\), and \(dx\) with \(\cos(\theta) \, d\theta\). The integral becomes \(\int \sin^{3}(\theta) \cos(\theta) \cdot \cos(\theta) \, d\theta = \int \sin^{3}(\theta) \cos^{2}(\theta) \, d\theta\).

Simplify the integral to \(\int \sin^{3}(\theta) \cos^{2}(\theta) \, d\theta\). Consider using a substitution such as \(u = \cos(\theta)\) or expressing powers of sine in terms of cosine to facilitate integration.

After integrating with respect to \(\theta\), substitute back \(\theta = \arcsin(x)\) to express the answer in terms of \(x\).

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주요 개념

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Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² - x²), √(a² + x²), or √(x² - a²). By substituting x with a trigonometric function (e.g., x = a sin θ), the radical expression transforms into a trigonometric identity, making the integral easier to evaluate.

Integral of Powers of x

Understanding how to integrate powers of x, such as x³, is essential. When combined with substitution, recognizing how to express x and dx in terms of θ allows the integral to be rewritten entirely in trigonometric terms, facilitating the integration process.

Trigonometric Identities

Familiarity with fundamental trigonometric identities, like sin²θ + cos²θ = 1, is crucial. These identities help simplify the integrand after substitution, enabling the integral to be expressed in a solvable form involving basic trigonometric functions.

Evaluate ∫ x³ √(1 - x²) dx using:c. A trigonometric substitution.