Page 90 - DMTH202_BASIC_MATHEMATICS_II

Unit 6: Properties of Definite Integral

The purpose of the formula is enclosed in the product rule for derivatives: Notes
d
´
F x G x     F x G x ´    
    F x G x
dx
which holds at any point where both functions are differentiable.
!
Caution One must provide strong enough hypotheses that the function F(x)G(x) is an
indefinite integral for the functioning the sense required for our integral.

6.1.6 Change of Variable
The change of variable formula (i.e., integration by substitution):
b g   b
a   f g t g ´ g   a  f   x dx .
    t dt 
The meaning of the formula is enclosed in the following statement which includes a sufficient
condition that permits this formula to be proved: Let I be an interval and g :[a, b]  I a continuously
   
differentiable function. Let that F : I  R is an integrable function. Then the function  g t g t
F
´
is integrable on [a, b] and the function f is integrable on the interval [g(a),g(b)] (or rather on
[g(a),g(b)] if g(b) < g(a)) and the identity holds. There are diverse assumptions under which this
might be applicable.
The proof is an application of the chain rule for the derivative of a composite function:
d
  
´
F G x  ´ F G    .x G x
dx

2
 cos x
Task Show that the integral  dx exists and use a change of variable to determine
0 x
the exact value.
6.1.7 Derivative of the Definite Integral

d x
What is a  ff   t dt ?
dx
x d x
We know that a  f   t dt is an indefinite integral of f and so, by definition, dx a  f   t dt  f   x at
all but finitely many points in the interval (a, b) if f is integrable on [a, b].
If we require to know more than that then there is the following edition which we have already
proved:
d x f   t dt  f   x at all points a < x < b at which f is continuous.
dx a 

!

Caution We should keep in mind, however, that there may also be various points where f
is discontinuous and so far the derivative formula holds.
d x
If we go ahead of the calculus interval, then the same formula is valid a  f   t dt  f   x but
dx
there may be several more than finitely many exceptions probable. For “most” values of t this
is true but there may even be infinitely many exceptions probable. It will still be true at points

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