TABLE OF CONTENTS The rule for differentiation of a product and the chain rule lead to theorems on integration by parts and substitution.
If F and G are continuous on [a, b], F'= f and G ?= g on [a, b] except on a finite set [9] then
| (15.42) |  |
We have ( FG)'= Fg+ Gf except on a finite set, the function FG is continuous on [a, b] and by the Fundamental Theorem, Equation (15.42) follows.
Provided that one of the integrals in Equation (15.43) below exists we obtain useful formula
| (15.43) |  |
The above formula is often rewritten in the easily remembered form
| (15.44) |  |
The assumptions of the theorem alone do not guarantee the existence of either integral in Equation (15.43) as the following example shows. For F( x)= x 2 sin x ?4, G( x)= x 2 cos x ?4 for x ?0 and F(0)= G(0)=0 neither 
nor 
exists because then both would exist by Equation (15.43) and so would their difference. However F( x) g( x) ? G( x) f( x)=4 x ?1 is not integrable. For F and G chosen as above, Equation (15.42) holds but Formula (15.43) does not make sense. Moreover this example also shows that the product of an integrable function
