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If f(x) and g(x) are continuous on [a, b], which one of the following statements is false?

A) the integral from a to b of the sum of f of x and g of x, dx equals the integral from a to b of f of x, dx plus the integral from a to b of g of x dx
B) the integral from a to b of the product of f of x and g of x, dx equals the integral from a to b of f of x, dx times the integral from a to b of g of x dx
C) the integral from a to b of 6 times f of x, dx equals 6 times the integral from a to b of f of x, dx
D) None are false

Respuesta :

abidemiokin

Answer: B) the integral from a to b of the product of f of x and g of x, dx equals the integral from a to b of f of x, dx times the integral from a to b of g of x dx

Step-by-step explanation:

The integral from a to b of the product of f of x and g of x, dx equals the integral from a to b of f of x, dx times the integral from a to b of g of x dx is FALSE.

The integral of product of two functions is not equal to the product of the integral of its individual function rather, the integral of product of function is solved using "integration by part method". This is done by assigning variables to each functions and applying the method to solve it. For example given the integral

Integral {f(x)g(x)}dx, we will assign any variable let's say 'u' to f(x) and 'dv' to g(x)dx

u = f(x); du/dx = f'(x)

du = f'(x)dx

dv = g(x)dx... (2)

Integrating both sides of eqn 2, we will have;

v = integral g(x)dx

Generally, Integral {udv}= uv - integral {vdu}... (3)

Substituting all variables into equation 3 we have;

Integral {f(x)g(x)}dx = f(x)g(x) - integral {integral g(x)dx}f'(x)dx

The given functions f(x) and g(x) are continuous on [a,b]. Option B "the integral from a to b of the product of f(x) and g(x), dx equals the integral from a to b of f(x), dx times the integral from a to b of g(x), dx" is the incorrect option.

Given information:

Functions f(x) and g(x) are continuous on [a,b].

Now, as the functions are continuous, their integral can be found in this domain.

Some important rules regarding the integral are:

  • Integral of a constant multiplied by a function is equal to constant multiplied by integral of the function. Constant could be taken out of the integral.
  • Integral of function f(x) plus g(x) is equal to sum of integral of individual functions.
  • Integral of product of two function can be found by integral by part rule. It is not equal to product of integral of individual functions.

From the above points, it can be concluded that option A and C are correct as they follow the rules. But the option B doesn't follow the rule and hence, its incorrect.

Therefore, option B "the integral from a to b of the product of f(x) and g(x), dx equals the integral from a to b of f(x), dx times the integral from a to b of g(x), dx" is the incorrect option.

Learn more about Definite Integration here: https://brainly.com/question/3486176

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