True or False
If F(x) and G(x) are antiderivatives of f(x), then F(x)=G(x)+C
If f'(x) = g(x) then integral g(x) dx = f(x) + C
Integral f(x) * g(x)dx = integral f(x)dx * integral g(x)dx
I have a feeling it's
False
True
True
As far as I know the first one is true, they differ by a constant.
yes, true
say f = x^2
g = x
then
F = (1/3) x^3 + C1
G = (1/2) x^2 + C2
integral f g dx = (1/3) x^4 + C
forget the constants for now, no need
F G = (1/6) x^5
like no way :)
You are partially correct. Let's go through each statement one by one:
Statement 1: If F(x) and G(x) are antiderivatives of f(x), then F(x) = G(x) + C
The statement is true. According to the Fundamental Theorem of Calculus, if F(x) is any antiderivative of f(x), then any other antiderivative G(x) of f(x) can be expressed as G(x) = F(x) + C, where C is the constant of integration.
Statement 2: If f'(x) = g(x), then integral g(x) dx = f(x) + C
The statement is true. This is a direct consequence of the Fundamental Theorem of Calculus. If f'(x) = g(x), then taking the integral of g(x) with respect to x gives us the original function f(x) up to a constant of integration. Therefore, integral g(x) dx = f(x) + C is correct.
Statement 3: Integral f(x) * g(x)dx = integral f(x)dx * integral g(x)dx
The statement is false. The product of two integrals is not equal to the integral of the product. Instead, the correct statement would be:
Integral f(x) * g(x) dx = integral (f(x) * g'(x)) dx
This is known as the integration by parts formula in calculus and involves integrating the product of two functions using a specific method.
So, in summary:
Statement 1: True
Statement 2: True
Statement 3: False