1

DEFINITE INTEGRAL

We know that the indefinite integral of a function f(x) is not unique and it varies by a

constant, known as the constant of integration i.e. I = f(x)dx F(x) c … (1)

where F(x) is the anti-derivative of f(x) and c is the constant of integration.

In geometrical and other applications of integration, it becomes necessary to find the

difference in the values of (1) for two different assigned values of the independent

variable x, say a and b. This difference is called the definite integral of f(x) over the

interval (a, b) and is denoted by

b

a

f(x)dx . Thus

bb

aa

f(x)dx F(x) c F(b) F(a) ..(2)

a is called the lower and b, the upper limit of integration; the value of

b

a

f(x)dx is unique.

Geometrical Interpretation of the Definite Integral

The

b

a

f(x)dx represents the

algebraic sum of the areas of

the regions bounded by the

graph of the function y = f(x),

the x-axis and the straight lines

x = a and x = b. The areas

above the x-axis enter into this

sum with plus sign, while those

below the x-axis enter it with a

minus sign.

f(x)

xx = bx = a

- -

-

++

The definite integral can be evaluated by using various techniques of integration studied

in indefinite integrals or by using the properties of definite integrals or by writing it as a

limit of a sum. Definite Integral as the Limit of a Sum

An alternative way of describing

b

a

f(x)dx is that the definite integral

b

a

f(x)dx is a

limiting case of the summation of an infinite series, provided f(x) is continuous on [a, b]

Note: The method to evaluate the integral, as limit of the sum of an infinite series is

known as Integration by First Principle.

Method to express the infinite series as definite integral:

Express the given series in the form 1 r

fn n

.

Then the limit is its sum whenn , i.e. n

1 rLim f

n n.

2

Replace r

n by x and

1

n by dx and

nLim by the sign of .

The lower and the upper limit of integration are the limiting values of r

n for the first and

the last term of r respectively.

Some particular cases of the above are

(a) n

n r 1

1 rlim f

n n or

1n 1

n r 1 0

1 rlim f f(x)dx

n n

(b)

pn

n r 1

1 rlim f f(x)dx

n n where

n

rLim 0

n (as r = 1)

and n

rLim p

n (as r = pn)

Fundamental Theorem of Integral Calculus:

Let F(x) be an anti-derivative of a continuous function f(x) on [a, b]. Then

b

a

f(x)dx F(b) F(a) .

Properties of Definite Integral

1.

a

a

f(x)dx 0 .

2. Change of variable of integration is immaterial so long as limits of integration remain the

same i.e.

b b

a a

f(x)dx f(t)dt .

3.

b a

a b

f(x)dx f(t)dx .

4.

b c b

a a c

f(x)dx f(x)dx f(x)dx

where the point c may lie between a and b or it may be exterior to (a, b).

Note: (i) This property is useful when f (x) is not continuous in [a, b], because we can break up

the integral into several integrals at the points of discontinuity so that the function is

continuous in the subintervals.

(ii) This property is true even when c lies outside the interval [a, b].

5.

b b

a a

f(x)dx f(a b x)dx

6.

a a / 2

0 0

f(x)dx [f(x) f(a x)]dx

Special cases:

3

If f (x) = f(a − x), then

a

a 2

0 0

f(x)dx 2 f(x)dx and If f (x) = − f(a − x), then

a

0

f(x)dx 0 .

7.

a a

a 0

f(x)dx [f(x) f( x)]dx

Special case:

aa

0a

2 f(x)dx, if f(x) is evenf(x)dx

0 , if f(x) is odd

8. If f (x) is a periodic function with period T, then

(a)

a nT T

a 0

f(x)dx n f(x)dx , where n I.

(i) In particular, if a = 0

nT T

0 0

f(x)dx n f(x)dx where n I .

(ii) If n = 1

a T T

a 0

f(x)dx f(x)dx .

(b)

nT T

mT 0

f(x)dx (n m) f(x)dx , where m, n I.

(c)

b nT b

a nT a

f(x)dx f(x)dx , where n I.

DIFFERENTIATION UNDER THE INTEGRAL SIGN

A. Leibnitz’s Rule

If g is continuous on [a, b] and f1 (x) and f2 (x) are differentiable functions whose values lie in

[a, b], then 2

1

f (x)

2 2 1 1

f (x)

dg(t)dt g(f (x))f (x) g(f (x))f (x)

dx.

In particular

x

0

dg t dt g x

dx.

B. If F(t) =

b

a

g(x,t)dx , then

b

a

dF g(x, t)dx

dt t, where

g

t represents the derivative of g

with respect to t keeping x constant.

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Inequalities

Sometimes you are asked to prove inequalities involving definite integrals or to estimate the

upper and lower bounds of a definite integral, where the exact value of the definite integral is

difficult to find. Under these circumstances, we use the following results:

(i)

b b

a a

f(x)dx | f(x) |dx Equality sign holds when f (x) is entirely of the same sign on [a, b].

(ii) If f(x) g(x) on [a, b], then

b b

a a

f(x)dx g(x)dx . In particular, if f(x) 0, then

b

a

f(x)dx 0 .

(iii) For a given function f (x) continuous on [a, b] if we are able to find two continuous

functions f1(x) and f2(x) on [a, b] such that f1(x) f(x) f2(x) x [a, b], then

b b b

1 2

a a a

f (x)dx f(x)dx f (x)dx .

(iv) If m and M are respectively the global minimum and global maximum of f (x) in [a, b]

then m (b a)

b

a

f(x)dx M(b a) .

(v)

b b b2 2

a a a

f(x)g(x)dx f (x)dx g (x) dx , where f(x) and g(x) are two integrable

functions.

Piecewise Continuous Functions

Consider a function f(x) defined on [a,b] which has discontinuities at finite number of points say

x1, x2, ....,xn. In such a case break the interval [a,b] into sub intervals such that f(x) is continuous

in each of the interval. Such a function is known as PIECEWISE continuous function. While

calculating the definite integral of such a function we have to break the interval [a,b] into sub

intervals and calculate the integral separately in each of the sub intervals and add all of this to

get the required answer.