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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||= 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
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