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
To determine which statement is false, let's evaluate each one.
Statement 1:
The integral from a to b of the sum of f(x) and g(x), dx equals the integral from a to b of f(x), dx plus the integral from a to b of g(x) dx.
This statement is true and follows from the linearity property of integrals. The integral of a sum of functions is equal to the sum of the integrals of each function.
Statement 2:
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.
This statement is false. The integral of the product of two functions is not generally equal to the product of their integrals. This is known as the multiplication rule for integrals, which states that the integral of the product of two functions is not equal to the product of their integrals unless the functions are constant or certain special conditions are met.
Statement 3:
The integral from a to b of 6 times f(x), dx equals 6 times the integral from a to b of f(x), dx.
This statement is true. It follows from the constant multiple rule for integrals, which states that the integral of a constant times a function is equal to the constant times the integral of the function.
Therefore, the false statement is statement 2: The integral from a to b of the product of f(x) and g(x), dx does not equal the integral from a to b of f(x), dx times the integral from a to b of g(x) dx.
Hence, the answer is: Statement 2 is false.
The false statement is:
"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"
This statement is not true in general. The product of the integrals of two functions is not equal to the integral of their product.
The first option I believe is false.
Hoped it helped!