The natural rules for the definite integral of sums and constant multiplies of functions, i.e.
|(f (x) + g(x))dx||= f (x)dx + g(x)dx|
|cf (x)dx||= c f (x)dx|
follow (by the Fundamental Theorem of Calculus) from the similar rules for antiderivatives, as we know prove.
Let F(x) and G(x) be two functions with F'(x) = f (x) , G'(x) = g(x) . We know by the addition rule for derivatives that
|F(x) + G(x) = [F(x) + G(x)]|
Writing this in terms of f and g yields
|f (x) + g(x) = [ f (x)dx + g(x)dx]|
As functions of b , the left and right hand sides of @@the sum rule@@ are antiderivatives of the two expressions above, so they differ by a constant. This constant must be zero, however, since the integrals are equal (both zero) for b = a , and the sum rule is proved.
Similarly, if c is a constant, we know that
|c F(x) = [cF(x)]|
|cf (x) = [c f (x)dx]|
As before, the @@constant multiple rule@@ asserts the equality of antiderivatives of these two expressions that agree for one value of b . Therefore the antiderivatives are equal, and the rule follows.