PROOF OF VARIOUS INTEGRAL FACTS/FORMULAS/PROPERTIES

In this section we've found the proof of several of the properties we saw in the Integrals section and also a couple from the applications of Integrals section.

Proof of: ∫k f(x) dx = k ∫f(x) dx here k is any numer

It is a very simple proof. Assume that F(x) is an anti-derivative of f(x) that is F′(x) = f(x). Then by the fundamental properties of derivatives we also have,

(k F(x))' = kF'(x) = k f(x)

and therefore k F(x) is an anti-derivative of k f(x) that is  (k F(x))' = k f(x). Though,

∫k f(x) dx = k F(x) + c = k ∫f(x) dx