Use any method to evaluate the integrals in Exercises 55–66.
∫ x² √(1 - x²) dx

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Recognize that the integral involves the expression \(\sqrt{1 - x^2}\), which suggests a trigonometric substitution because \(1 - x^2\) resembles the identity \(\sin^2\theta + \cos^2\theta = 1\).

Use the substitution \(x = \sin\theta\), which implies \(dx = \cos\theta \, d\theta\). This transforms the integral into terms of \(\theta\).

Rewrite the integral \(\int x^2 \sqrt{1 - x^2} \, dx\) as \(\int \sin^2\theta \sqrt{1 - \sin^2\theta} \cos\theta \, d\theta\). Since \(\sqrt{1 - \sin^2\theta} = \cos\theta\), the integral becomes \(\int \sin^2\theta \cos\theta \cos\theta \, d\theta = \int \sin^2\theta \cos^2\theta \, d\theta\).

Simplify the integral to \(\int \sin^2\theta \cos^2\theta \, d\theta\). Use trigonometric identities such as the double-angle formulas to express powers of sine and cosine in terms of cosines of multiple angles, which makes the integral easier to evaluate.

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

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Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral, the integral can often be transformed into a more manageable form. This method is especially useful when the integral contains composite functions like √(1 - x²).

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Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots with trigonometric functions to simplify integration. For integrals containing √(1 - x²), substituting x = sin(θ) leverages the identity 1 - sin²(θ) = cos²(θ), making the integral easier to evaluate.

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Understanding how to integrate polynomial functions like x² is fundamental. Polynomials are integrated by increasing the exponent by one and dividing by the new exponent. This basic skill is often combined with substitution methods to solve more complex integrals.

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