Page 92 - DMTH202_BASIC_MATHEMATICS_II

Unit 6: Properties of Definite Integral

15. Complete the following: Notes
b b

...............................=  r a  f   x dx s  a  g    ,x dx r s .
6.2 Summary

 There are certain properties of definite integral which makes the integration simpler.

 The fundamental properties of integrals are simply attained for us since the integral is
defined straightforwardly by differentiation.
 When a function involves a calculus integral on an interval it must also contain a calculus
integral on all subintervals.

 When a function involves a calculus integral on two contiguous intervals it must also
contain a calculus integral on the amalgamation of the two intervals.
b b
 The formula for inequalities is a  f   x dx  a  g   x dx if f (x)  g(x) for all but finitely several
points x in (a,b).
 If both functions f (x) and g(x) contain a calculus integral on the interval [a,b] then any
linear combination r f (x)+s g(x) (r, s  R) also encloses a calculus integral on the interval
[a,b] and, furthermore, the identity is ought to be hold.

 Integration by parts formula is the purpose of the formula is
b b
    x dx 
a  F x G ´ F x G a  ´ F x G
    x 
    x dx enclosed in the product rule for derivatives:
d
    x 
    
F x G x F x G ´ ´ F x G
    x which holds at any point where both functions are
dx
differentiable.
 The first two are similar to the properties of limits. The other two are very perceptive and
relate to the notion of area.
6.3 Keywords
Change of Variable: The change of variable formula (i.e., integration by substitution):

b g   b
    t dt 
a   f g t g ´ g   a  f   x dx .
Fundamental Theorem of Calculus: This theorem converts the integral from a mathematical
inquisitiveness to a prevailing tool that is accessed in science, engineering, economics and many
other areas.

6.4 Review Questions

1. Supply the details needed to prove the integration by parts formula in the special case
where F and G are continuously differentiable everywhere.
2. Supply the details needed to prove the change of variable formula in the special case
where F and G are differentiable everywhere.

3. Let F(x) =|x| and G(x)= x 2 sin , G (0) = 0. Does
1

1
0
0   ´ F G x G ´ F G   F G     sin 1 ?
1
    x dx 

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