Multiple Choice If ∫ 3 ^ 7 f ( x ) d x = 5 and ∫ 3 ^ 7 g ( x ) d x = 3 , then all of the f

step 1 We're given that the integral from 3 to 7 of F of X is 5, and the integral from 3 to 7 of G of X

Multiple Choice If ∫ 3 ^ 7 f ( x ) d x = 5 and ∫ 3 ^ 7 ...

Multiple Choice If $\int _ { 3 } ^ { 7 } f ( x ) d x = 5$ and $\int _ { 3 } ^ { 7 } g ( x ) d x = 3 ,$ then all of the following must be true except (A) $$\int _ { 3 } ^ { 7 } f ( x ) g ( x ) d x = 15$$ (B) $$\int _ { 3 } ^ { 7 } [ f ( x ) + g ( x ) ] d x = 8$$ (C) $$\int _ { 3 } ^ { 7 } 2 f ( x ) d x = 10$$ (D) $$\int _ { 3 } ^ { 7 } [ f ( x ) - g ( x ) ] d x = 2$$ (E) $$\int _ { 7 } ^ { 3 } [ g ( x ) - f ( x ) ] d x = 2$$

Multiple Choice If $\int _ { 3 } ^ { 7 } f ( x ) d x = 5$ and $\int _ { 3 } ^ { 7 } g ( x ) d x = 3 ,$ then
all of the following must be true except
(A) $$\int _ { 3 } ^ { 7 } f ( x ) g ( x ) d x = 15$$
(B) $$\int _ { 3 } ^ { 7 } [ f ( x ) + g ( x ) ] d x = 8$$
(C) $$\int _ { 3 } ^ { 7 } 2 f ( x ) d x = 10$$
(D) $$\int _ { 3 } ^ { 7 } [ f ( x ) - g ( x ) ] d x = 2$$
(E) $$\int _ { 7 } ^ { 3 } [ g ( x ) - f ( x ) ] d x = 2$$

Key Concepts

Linearity of Definite Integrals

This principle states that the integral of a sum of functions is equal to the sum of their integrals, and similarly, the integral of a difference of functions is equal to the difference of their integrals. This property is critical when breaking down more complex integrals into simpler parts to evaluate the overall integral.

Constant Factor Rule

The constant factor rule asserts that a constant multiplied by a function can be taken outside the integral, meaning that the integral of a constant times a function is equal to the constant multiplied by the integral of the function. This is essential for simplifying the evaluation of integrals where constants are involved.

Reversal of Limits in Definite Integrals

This property indicates that reversing the limits of integration changes the sign of the integral. It is used to handle cases where the integration is performed in the reverse order relative to the original, requiring an adjustment of the sign of the result.

Nonlinearity of the Integral with Respect to Multiplication

Unlike sums and differences, the integral of a product of two functions is not generally equal to the product of their individual integrals. This important concept prevents incorrect simplifications when dealing with the integration of multiplied functions and highlights a key distinction in the properties of integration.

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