Question

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

~the integral from a to b of the difference of f of x and g of x, dx equals the integral from a to b of f of x, dx minus the integral from a to b of g of x dx

~the integral from a to a of f of x, dx equals 2 times the integral from 0 to a of f 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
(MY ANSWER)

~the integral from a to b of 3 plus f of x, dx equals 3 plus the integral from a to b of f of x, dx

Answer 1

nope. sums and differences work that way, but not products. Consider

f(x) = x
g(x) = x
∫f * ∫g = 1/2 x^2 * 1/2 x^2 = 1/4 x^4
∫fg = ∫x^2 = 1/3 x^3

∫f-g = ∫f - ∫g TRUE


∫[a,a] f(x) dx = 0
NOT 2∫[0,a]

∫3+f = ∫3 + ∫f
NOT 3+∫f