Proof of: ∫ f(x) + g(x) dx = ∫ f(x) dx + ∫g(x) dx

It is also a very easy proof. Assume that F(x) is an anti-derivative of f(x) and that G(x) is an anti-derivative of g(x). Therefore we have that F′(x) = f(x) and G′(x) = g(x).

Fundamental properties of derivatives also give us that

(F(x) + G(x))' = F'(x) + G(x) = f(x) + g(x)

and thus F(x) + G(x) is an anti-derivative of f(x) + g(x) and F(x) - G(x) is an anti- derivative of f(x)- g(x). So,

∫ f(x) + g(x) dx = F(x) + G(x) + c =∫ f(x) dx + ∫g(x) dx