Computing Integrals
Addition and Multiplication by a Constant
The natural rules for the definite integral of sums and constant multiplies of functions, i.e.
sumrule, constmult
 (f (x) + g(x))dx
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=
f (x)dx +
g(x)dx
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||
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cf (x)dx
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= c
f (x)dx
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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)]
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Writing this in terms of f and g yields
f (x) + g(x) = [
f (x)dx +
g(x)dx]
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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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c
F(x) = [cF(x)]
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or
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cf (x) = [c
f (x)dx]
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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.
