Page 70 - DMTH202_BASIC_MATHEMATICS_II

Unit 4: Definite Integral

Notes
k

1 1 x  
 tan   
1 1 2

2 2 0 
k
1

 2tan 2x  
0  2

1

0
 2tan 2k  
2
 
1

 2tan 2k   2k  tan
4 4
1
k
1
 2k   
2
Task Evaluate the following integrals:
3
dx
1. 2 
2 x  1
1 2
1 x

2.  2 dx

0 1 x
Self Assessment
Fill in the blanks:
11. .................................. states that if f(x) is a continuous function defined on closed interval
[a, b] and F(x) is integral of f(x).
b b
x
b
a

x
x
12.  f ( )dx  F ( ),then  f ( )dx  ( F  ) x  F ( ) F ( ) .................................. .
a a
13. In Fundamental theorem of integral calculus, a is called lower limit, b is called upper limit
and F(b) – F(a) is called the..................................of the definite integral.
14. The Fundamental theorem of integral calculus is very useful as it gives us a method of
calculating the definite integral more easily without calculating the ................................... .
15. The crucial operation in evaluating a definite integral is that of finding a function whose
derivative is equal to the .................................. .
4.3 Summary

 The Definite Integral comprises extensive number of applications in mathematics, the
physical sciences and engineering.

 Let f(x) be a continuous real valued function defined on the closed interval [a, b]. Divide
the interval [a, b] into n equal parts each of width h by points a + h, a + 2h, a + 3h, …, a +
(n – 1) h.

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