Question

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) $$\int _ { 7 } ^ { 3 } [ g ( x ) - f ( x ) ] d x = 2$$

Short answer

(A) \(\int _ { 3 } ^ { 7 } f ( x ) g ( x ) dx = 15\) and (E) \(\int _ { 7 } ^ { 3 } [ g ( x ) - f ( x ) ] dx = 2\) are not necessarily true according to the properties of integrals. Therefore, the correct answer is both A and E.

Step-by-step solution

Analyze Options

First, it's important to understand what each option implies about the rules and properties of integrals:\n (A) Implies the product of the functions under the integral sign which is not necessarily equivalent to the product of the two given integrals. Thus, it is not generally true.\n (B) According to the property of integrals, the integral of the sum of two functions is equal to the sum of their integrals.\n (C) Implies that the constant 2 multiples the integral, given the property of integrals, the integral of constant times function equals the constant times the integral of the function.\n (D) The property of integrals states that the integral of the difference of two functions is the difference of the integral of the two functions. \n (E) Notes that the integral limits are reversed. The property of integrals states that the integral from a to b of a function is the negative of the integral from b to a of that function.

Verify each Option

Now that we understand what each option is saying about the properties of integrals, let's substitute the values from the original problem into the equations: (A) \(\int_ { 3 } ^ { 7 } f ( x ) g ( x ) d x \) does not necessarily equal 15. This is not a property of integrals and hence, it's false. (B) \(\int_ { 3 } ^ { 7 } [ f( x ) + g( x ) ] dx \) equals \(\int_ { 3 } ^ { 7 } f( x ) dx + \int_ { 3 } ^ { 7 } g( x ) dx \), which equals 5 + 3 = 8. It's true. (C) \(\int_ { 3 } ^ { 7 } 2f ( x ) dx \) equals \(2 * \int_ { 3 } ^ { 7 } f( x ) dx \), which equals 2*5 = 10. It's true. (D) \(\int_ { 3 } ^ { 7 } [ f( x ) - g( x ) ] dx \) equals \(\int_ { 3 } ^ { 7 } f( x ) dx - \int_ { 3 } ^ { 7 } g( x ) dx \), which equals 5 - 3 = 2. It's true. (E) \(\int_ { 7 } ^ { 3 } [ g( x ) - f( x ) ] dx \) equals \(-\int_ { 3 } ^ { 7 } [ f( x ) - g( x ) ] dx \), which equals -2. It's false as its value needs to be 2.