Page 62 - DMTH202_BASIC_MATHEMATICS_II

Unit 4: Definite Integral

2 n (n  1)(2n  1) Notes
2
2
2
9. 1  2  3   (n  1)  ................................
6
 2
( n n 
3 
1)
3
3
3
10. 1  2  3    (n 1)    ................................
 2 
4.2 Fundamental Theorem of Integral Calculus
Fundamental theorem of integral calculus states that if f(x) is a continuous function defined on
closed interval [a,b] and F(x) is integral of f(x) i.e., f ( )dx  F ( ),then  f ( )dx  F ( ),then a is
x
x

x
x
called lower limit, b is called upper limit and F(b) – F(a) is called the value of the definite integral
and is always unique.
Notes This theorem is very useful as it gives us a method of calculating the definite
integral more easily, without calculating the limit of a sum.

Did u know? The crucial operation in evaluating a definite integral is that of finding a
function whose derivative is equal to the integrand.
This toughens the relationship among differentiation and integration.

Example: Evaluate the following integrals:
3
3
(i)  (2x  1) dx
1
2 1
(ii)  dx
 1 3x  2
 /2
2
(iii)  sin xdx
0
 /4
(iv)  tanxdx
0
 /4
(v)  secxdx
0
1 1
(vi)  dx
0 1  x 2
1 dx
(vii)  2

0 1 x
3 x
(viii)  2 dx
2 x  1
Solution:
3 (2x  1)  3 1 4  3 1  4 1 
4
4
3
(i) I   (2x  1) dx  4 2   8 2x   1   8   (7)  (3) 8 (2320)  290


1 1  1  
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