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The natural rules for the definite integral of sums and constant multiplies of functions, i.e.

sumrule, constmult

(f (x) + g(x))dx = f (x)dx + g(x)dx  
cf (x)dx = cf (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

cF(x) = [cF(x)]    

or

cf (x) = [cf (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.

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