Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.
(b) ∫₃⁶ (―3g(𝓍)) d𝓍

Verified step by step guidance

1. Step 1: Recall the property of integrals that allows you to factor out constants. Specifically, for any constant c and function g(x), ∫ₐᵇ c·g(x) dx = c·∫ₐᵇ g(x) dx.
2. Step 2: Apply this property to the given integral ∫₃⁶ (―3g(𝓍)) d𝓍. Here, the constant is ―3, so the integral becomes ―3·∫₃⁶ g(𝓍) d𝓍.
3. Step 3: Substitute the value of ∫₃⁶ g(𝓍) d𝓍, which is provided as 1, into the expression from Step 2.
4. Step 4: Multiply the constant ―3 by the value of the integral ∫₃⁶ g(𝓍) d𝓍 to simplify the expression.
5. Step 5: The result of the multiplication gives the value of the integral ∫₃⁶ (―3g(𝓍)) d𝓍. Ensure you understand how the constant factor affects the integral.

Key Concepts

Properties of Integrals

The properties of integrals, particularly the linearity property, state that the integral of a sum of functions is the sum of their integrals, and that a constant can be factored out of an integral. This means that for any constant 'c' and function 'f(x)', ∫c f(x) dx = c ∫f(x) dx. Understanding these properties is essential for simplifying and evaluating integrals.

Definite Integrals

Definite integrals represent the signed area under a curve between two limits. The notation ∫ₐᵇ f(x) dx indicates the integral of f(x) from 'a' to 'b'. The value of a definite integral can be interpreted as the accumulation of quantities, and it can be positive, negative, or zero depending on the function's behavior over the interval.

Substitution in Integrals

Substitution is a technique used in integration to simplify the process by changing the variable of integration. This method often involves setting u = g(x) for some function g, which transforms the integral into a more manageable form. Understanding how to apply substitution effectively can greatly aid in evaluating complex integrals.