7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
30. ∫ x³√(1 - x²) dx

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Step 1: Recognize that the integral involves a square root of the form √(1 - x²), which suggests using the trigonometric substitution x = sin(θ). This substitution simplifies √(1 - x²) to cos(θ).

Step 2: Substitute x = sin(θ) into the integral. Compute dx = cos(θ) dθ and replace x³ with (sin(θ))³. The integral becomes ∫ (sin(θ))³ cos(θ) cos(θ) dθ, which simplifies to ∫ (sin(θ))³ (cos(θ))² dθ.

Step 3: Simplify the integral further by using the trigonometric identity (cos(θ))² = 1 - (sin(θ))². Replace (cos(θ))² in the integral, resulting in ∫ (sin(θ))³ [1 - (sin(θ))²] dθ.

Step 4: Expand the expression inside the integral to separate terms. The integral becomes ∫ (sin(θ))³ dθ - ∫ (sin(θ))⁵ dθ. These integrals can be solved using standard trigonometric integration techniques.

Step 5: After solving the integral in terms of θ, convert back to the original variable x using the substitution x = sin(θ). Use the relationship θ = arcsin(x) and simplify the result to express the solution in terms of x.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of expressions like 1 - x². By substituting x with a trigonometric function (e.g., x = sin(θ) or x = cos(θ)), the integral can often be transformed into a more manageable form, allowing for easier evaluation.

Integral of a Polynomial

The integral of a polynomial function involves finding the antiderivative of the polynomial expression. For example, the integral of x³ is (1/4)x⁴ + C, where C is the constant of integration. Understanding how to integrate polynomials is essential for solving integrals that include polynomial terms, such as x³ in the given problem.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is crucial when performing trigonometric substitutions, as it allows for the simplification of expressions involving square roots. In the context of the integral ∫ x³√(1 - x²) dx, this identity helps to express √(1 - x²) in terms of trigonometric functions, facilitating the integration process.

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