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

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