# 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 = 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)]

or

 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.