Page 93 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes 4. Supply the details needed to prove the change of variable formula in the special case
where G is strictly increasing and differentiable everywhere
5. Show that the function f (x) = x is integrable on [–1,2] and compute its definite integral there.
2
6. Show that each of the following functions is not integrable on the interval stated:
(a) f (x) = 1 for all x irrational and f (x) = 0 if x is rational, on any interval [a,b].
(b) f (x) = 1 for all x irrational and f (x) is undefined if x is rational, on any interval [a,b].
(c) f (x) = 1 for all x 6= 1,1/2,1/3,1/4, . . . and f (1/n) = cn for some sequence of positive
numbers c , c , c , . . . , on the interval [0,1].
1 2 3
7. Determine all values of p for which the integrals
1 p  p
0  x dx or 1  x dx

8. Are the following additivity formulas for infinite integrals valid:
 a b 
(a)   f   x dx    f   x dx  a  f   x dx  b  f   x dx ?
  n
(b)   f   x dx   n 1  f   x dx ?
n 1
  n
(c)   f   x dx   n 1   f   x dx ?
n
9. Evaluate the following integral using properties of definite integrals and interpreting
integrals as areas:
6
1  4x   2 dx

10. Evaluate the following integral using properties of definite integrals and interpreting
integrals as areas.

2  3 9  
2   5u  5u   du
  2 
Answers: Self Assessment

1. Definite 2. upper
3. function 4. differentiation

5. subintervals 6. contiguous
b
7. sum 8. a   f  g   x dx
9. inequalities 10. linear combination
11. parts 12. change of variable
c
13. indefinite 14. b  f   x dx
b
15. a    rf   x  sg   x   dx

88 LOVELY PROFESSIONAL UNIVERSITY