Key Concepts Definite

b
b
Definition: If f is continuous in (a, b) and bounded then  f (x) dx = F(x)
a
a
= F(b) – F(a) is

called the definite integral of f(x) between the limits a and b.

d
where F( x )   f ( x )
dx
Very Important:
b
1. If  f (x ) dx = 0, then the equation f (x) = 0 has atleast one root in (a, b) provided f is
a
continuous in (a, b).
Note that the converse is not true.
e.g. If 2a + 3b + 6c = 0 then the QE ax2 + bx + c = 0 must have a root in (0, 1)
b
2. (a) If f (x) > 0 in (a, b) then  f ( x ) dx > 0 provided a < b.
a

r
b b

(b)  f (x ) dx   | f (x ) | dx

3.
a
a

 f (x ) dx = 0 only if
a
a

f (a) is defined.
si
.B
0
dt
e.g.  0
0
t
G

b g 1 ( b )
4.  f ( x ) · dg( x ) =  f (x ) · g' (x ) dx .
1
[a and b are limit of g (x)]
a
@

g (a )
Note that when g (x) = b then x = g–1(b) and g(x) = a then x = g–1(a);
b
 d  b
5.   dx f (x)  dx  f (x)a if f(x) is continuous in (a, b) however if f(x) is discontinuous
a
b
 d  c b
at x = c  (a, b) then   f (x)  dx  f (x)a  f (x)  [convergent and divergent]
dx c
a 
b  b
6. Lim  f n (x)dx     Lim f n (x) dx ;

n     n  
a  a
7. If g (x) is the inverse of f (x) and f (x) has domain x  [a, b] where f (a) = c and
b d
f (b) = d then the value of  f (x) dx +  g(y) dy = (bd – ac)
a c

ETOOSINDIA www.etoosindia.com
Plot No. 46, In front of Skyline Apartments corner Building,
India’s No.1 Online Coaching
Rajeev Gandhi Nagar, Kota, Rajasthan
 Definite
8. Remember the values of the following def. integral
/2 /2 /2 / 2
2 2 
(a)  sin x dx =  cos x dx =1 (b)  sin x dx   cos x dx 
4
0 0 0 0
/2 /2 /2 /2
3 2 3 4 4 3
(c)  sin x dx   cos x dx  3 (d)  sin x dx   cos x dx 
16
0 0 0 0

PROPERTIES OF DEFINITE INTEGRAL

WITH ILLUSTRATIONS
(A) PROPERTIES:

b b b a

P1  f(x) dx =  f(t)dt ; P2  f(x) dx = –  f(x)dx

a a a b

b c b
P3  f(x) =  f(x) dx +  f(x) dx

r
a a c

provided f has a piece wise continuity when f is not uniformly defined in (a, b)

P4
a


a
f(x) dx =
a

 f (x)  f (x)  dx
0
si = [
0
a
if f (x) is odd

2 f (x)dx if f (x) is even

.B
0

b b a a
P5  f(x) dx =  f(a + b  x) dx or  f(x) dx =  f(a  x) dx
G

a a 0 0

2a a a 0 if f (2 a  x)   f (x)
P6 f(x) dx = f (x) d x + f (2a  x) d x  [ a
@

  
0 0 0 2  f (x) dx if f ( 2 a  x)  f (x)
0

na a

P7  f(x) dx = n  f(x)dx where f(a + x) = f(x) n  I

0 0

na a
Note:  f (x) dx  (n m) f (x) dx , f (x) is periodic with period = a (n, m  N, n > m)
ma 0

(B) DERIVATIVES OF ANTIDERIVATIVES (LEIBNITZ RULE)

If f is continuous and g(x) and h(x) are differentiable function.
h(x)
d
dx  f (t) dt = f ( h(x) )·h(x) – f (g(x) ). g(x) (integrand of a continuous function is
g(x)

always differentiable)
b
h(x) h(x)
d   
Note: dx  f (x, t) dt = f x, h(x) h'(x)  f x, g(x)g'(x)   f (x, t)  dt
x
g(x) g(x)  

ETOOSINDIA www.etoosindia.com
Plot No. 46, In front of Skyline Apartments corner Building,
India’s No.1 Online Coaching
Rajeev Gandhi Nagar, Kota, Rajasthan
 Definite

(C) DEFINITE INTEGRAL AS A LIMIT OF SUM

Fundamental theorem of integral calculus
b

 f (x) dx = Limit h [f (a) + f (a+h) + f (a + 2h)+ .... + f (a + n  1 h) ]

h0
a n
b n1
or  f (x) d x = Lim
h0
it h  f (a + r h) where b  a = nh
a n r0

(D) ESTIMATION OF DEFINITE INTEGRAL AND GENERAL INEQUALITIES

IN INTEGRATION:
Not all integers can be evaluated using the technique discussed so far.

(a) For a monotonic increasing function in (a, b)

b

(b – a) f(a) <  f (x) dx

a
< (b – a) f(b)

r
(b) Foa a monotonic decreasing function in (a, b)

f(b). (b – a) <
b

 f (x) dx
a
< (b – a) f(a)
si
.B
(c) For a non monotonic function in (a, b)
b

 f (x) dx
G

f(c) (b – a) < < (b – a) f(b)

a

(d) In addition to this note that

@

b b

 f (x) dx <  | f (x)| dx

a
equality holds when f (x) lies completely above the x-axis
a

(E) WALLI'S THEORM & REDUCTION FORMULA

/2
n [(n 1)(n  3)....1or 2 ] [ (m 1)(m  3)...1or 2]
 sin x cos m x dx = K
0 (m  n) (m  n  2)...1or 2
(m, n are non-negative integer)

(F) DIFFERENTIATING AND INTEGRATING SERIES

ETOOSINDIA www.etoosindia.com
Plot No. 46, In front of Skyline Apartments corner Building,
India’s No.1 Online Coaching
Rajeev Gandhi Nagar, Kota, Rajasthan