Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.

(a) ∫₁⁴ 3f(𝓍) d𝓍

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Step 1: Recall the property of integrals that allows constants to be factored out. Specifically, for any constant c and function ƒ(𝓍), ∫ₐᵇ cƒ(𝓍) d𝓍 = c∫ₐᵇ ƒ(𝓍) d𝓍.

Step 2: Apply this property to the given integral ∫₁⁴ 3ƒ(𝓍) d𝓍. Factor out the constant 3, so the integral becomes 3∫₁⁴ ƒ(𝓍) d𝓍.

Step 3: Notice that the integral ∫₁⁴ ƒ(𝓍) d𝓍 is not directly provided in the problem. However, you can calculate it using the additive property of integrals: ∫₁⁶ ƒ(𝓍) d𝓍 = ∫₁⁴ ƒ(𝓍) d𝓍 + ∫₄⁶ ƒ(𝓍) d𝓍.

Step 4: Substitute the known values into the equation. From the problem, ∫₁⁶ ƒ(𝓍) d𝓍 = 10 and ∫₄⁶ ƒ(𝓍) d𝓍 = 5. Solve for ∫₁⁴ ƒ(𝓍) d𝓍 by subtracting: ∫₁⁴ ƒ(𝓍) d𝓍 = 10 - 5.

Step 5: Replace ∫₁⁴ ƒ(𝓍) d𝓍 in the expression 3∫₁⁴ ƒ(𝓍) d𝓍 with the calculated value from Step 4. The final integral is now expressed as 3 multiplied by the result of ∫₁⁴ ƒ(𝓍) d𝓍.

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Properties of Definite Integrals

Definite integrals have several key properties, including linearity, which states that the integral of a sum of functions is the sum of their integrals. Additionally, the integral of a constant multiplied by a function can be factored out, allowing for simplification in calculations. Understanding these properties is essential for evaluating integrals efficiently.

Integration of Constant Multiples

When integrating a function multiplied by a constant, the constant can be factored out of the integral. For example, ∫ₐᵇ kƒ(𝓍) d𝓍 = k∫ₐᵇ ƒ(𝓍) d𝓍, where k is a constant. This property simplifies the evaluation of integrals, making it easier to compute the area under the curve of the function multiplied by a constant.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫ₐᵇ f(𝓍) d𝓍 = F(b) - F(a). This theorem is crucial for evaluating definite integrals and understanding the relationship between a function and its integral over a specified interval.

Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.(a) ∫₁⁴ 3f(𝓍) d𝓍