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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

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