If f(x) and g(x) are continuous on [a, b], which one of the following statements is false?
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.
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.
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.
None are false

Solution:

It is given that,

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.

So,

The integral of the product of two functions is not equal to the product of the integral of its individual function; rather, the integral of the product of function is solved using "integration by part method".

This is done by assigning variables to each function and applying the method to solve it.

Example given the integral

Integral {f(x)g(x)}dx, --- (1)

We will assign any variable let assume that '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 equation 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

Therefore, 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.

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

Summary:

If f(x) and g(x) are continuous on [a, 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 is false.