Use geometry and properties of integrals to evaluate

∫₀¹ (2𝓍 + √(1―𝓍²) + 1) d𝓍

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Step 1: Break the integral into separate terms using the property of integrals: ∫ₐᵇ [f(𝓍) + g(𝓍) + h(𝓍)] d𝓍 = ∫ₐᵇ f(𝓍) d𝓍 + ∫ₐᵇ g(𝓍) d𝓍 + ∫ₐᵇ h(𝓍) d𝓍. This gives ∫₀¹ (2𝓍 + √(1―𝓍²) + 1) d𝓍 = ∫₀¹ 2𝓍 d𝓍 + ∫₀¹ √(1―𝓍²) d𝓍 + ∫₀¹ 1 d𝓍.

Step 2: Evaluate ∫₀¹ 2𝓍 d𝓍. Use the power rule for integration: ∫ 𝓍ⁿ d𝓍 = (𝓍ⁿ⁺¹)/(n+1) + C. Here, n = 1, so ∫ 2𝓍 d𝓍 = 2 * (𝓍²/2) = 𝓍². Apply the limits of integration from 0 to 1.

Step 3: Evaluate ∫₀¹ √(1―𝓍²) d𝓍. Recognize that √(1―𝓍²) represents the equation of a semicircle with radius 1 centered at the origin. The integral ∫₀¹ √(1―𝓍²) d𝓍 calculates the area of one-quarter of the circle. Use the formula for the area of a circle, A = πr², and divide by 4.

Step 4: Evaluate ∫₀¹ 1 d𝓍. The integral of a constant c over [a, b] is given by c(b―a). Here, c = 1, a = 0, and b = 1, so ∫₀¹ 1 d𝓍 = 1 * (1―0).

Step 5: Combine the results from Steps 2, 3, and 4. Add the values obtained for ∫₀¹ 2𝓍 d𝓍, ∫₀¹ √(1―𝓍²) d𝓍, and ∫₀¹ 1 d𝓍 to get the final result of the integral.

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Definite Integrals

A definite integral represents the signed area under a curve between two points on the x-axis. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the limits of integration. The value of a definite integral can be interpreted geometrically as the accumulation of the area between the function f(x) and the x-axis from x = a to x = b.

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Properties of integrals, such as linearity, allow us to break down complex integrals into simpler parts. For example, ∫(f(x) + g(x)) dx = ∫f(x) dx + ∫g(x) dx. This property is particularly useful when evaluating integrals that consist of multiple terms, as it enables the evaluation of each term separately.

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Geometric Interpretation of Functions

Understanding the geometric interpretation of functions is crucial for evaluating integrals. For instance, the term √(1 - x²) represents a semicircle with radius 1. Recognizing the shapes formed by the functions involved can simplify the evaluation of the integral by allowing the use of geometric area formulas instead of purely algebraic methods.

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