INTEGRALS

INTEGRALS

7 Chapter

7.1 Overview

7.1.1

Let

d dx

F (x) =

f

(x). Then, we write

f

( x) dx = F

(x) +

C. These integrals are

called indefinite integrals or general integrals, C is called a constant of integration. All

these integrals differ by a constant.

7.1.2 If two functions differ by a constant, they have the same derivative.

7.1.3 Geometrically, the statement f ( x) dx = F (x) + C = y (say) represents a

family of curves. The different values of C correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself. Further, the tangents to the curves at the points of intersection of a line x = a with the curves are parallel.

7.1.4 Some properties of indefinite integrals

(i) The process of differentiation and integration are inverse of each other,

i.e.,

d dx

f

(

x) dx =

f

(x)

and

f '( x) dx = f ( x) + C ,

where

C

is

any

arbitrary constant.

(ii) Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. So if f and g are two functions such that

d dx

f

( x) dx

=

d dx

g(x) dx

, then

f

( x) dx

and

g ( x) dx are equivalent.

(iii) The integral of the sum of two functions equals the sum of the integrals of

the functions i.e., ( f ( x) + g ( x) ) dx = f ( x) dx + g ( x) dx .

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(iv) A constant factor may be written either before or after the integral sign, i.e.,

a f ( x) dx = a f ( x) dx , where `a' is a constant.

(v) Properties (iii) and (iv) can be generalised to a finite number of functions f1, f2, ..., fn and the real numbers, k1, k2, ..., kn giving

(k1 f1 ( x) + k2 f2 ( x) + ...+, kn fn ( x)) dx = k1 f1 ( x) dx + k2 f2 ( x) dx +... + kn fn ( x) dx

7.1.5 Methods of integration

There are some methods or techniques for finding the integral where we can not directly select the antiderivative of function f by reducing them into standard forms. Some of these methods are based on

1. Integration by substitution 2. Integration using partial fractions 3. Integration by parts.

7.1.6 Definite integral

b

The definite integral is denoted by f ( x) dx , where a is the lower limit of the integral a

and b is the upper limit of the integral. The definite integral is evaluated in the following two ways:

(i) The definite integral as the limit of the sum

(ii) 7.1.7

b

f ( x) dx = F(b) ? F(a), if F is an antiderivative of f (x).

a

The definite integral as the limit of the sum

b

The definite integral f ( x) dx is the area bounded by the curve y = f (x), the ordia

nates x = a, x = b and the x-axis and given by

b

a

f

( x) dx =

(b ? a)

lim

n

1 n

f

(a)

+

f

(a + h) +... f (a + (n ? 1) h)

INTEGRALS 145

or

b

f

( x) dx =

lim

h0

h

f

(a)

+

f

(a + h)

+... +

f

(a + (n ? 1) h)

,

a

7.1.8 (i)

(ii)

(iii)

where h = b ? a 0 as n . n

Fundamental Theorem of Calculus

Area function : The function A (x) denotes the area function and is given

x

by A (x) = f ( x) dx . a

First Fundamental Theorem of integral Calculus

Let f be a continuous function on the closed interval [a, b] and let A (x) be the area function . Then A (x) = f (x) for all x [a, b] .

Second Fundamental Theorem of Integral Calculus

Let f be continuous function defined on the closed interval [a, b] and F be an antiderivative of f.

7.1.9

b

f ( x) dx = [F( x)]ba = F(b) ? F(a).

a

Some properties of Definite Integrals

b

b

P0 : f ( x) dx = f (t ) dt

a

a

b

a

a

P1 : f ( x) dx = ? f ( x) dx , in particular, f ( x) dx = 0

a

b

a

b

c

b

P2 : f ( x) dx = f ( x) dx + f ( x) dx

a

a

c

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b

b

P3 : f ( x) dx = f (a + b ? x) dx

a

a

a

a

P4 : f ( x) dx = f (a ? x) dx

0

0

2a

a

a

P5 : f ( x) dx = f ( x) dx + f (2a ? x) dx

0

0

0

P6 :

2a

f

( x) dx

=

a 2

f

( x) dx,if

0

0

f (2a - x) = f (x) ,

0, if f (2a - x) = - f (x).

a

a

P7 : (i) f ( x) dx = 2 f ( x) dx , if f is an even function i.e., f (?x) = f (x)

?a

0

a

(ii) f ( x) dx = 0, if f is an odd function i.e., f (?x) = ?f (x) ?a

7.2 Solved Examples

Short Answer (S.A.)

Example

1 Integrate

2a x

?

b x2

+ 3c 3

x2

w.r.t.

x

Solution

2a x

b ?

x2

+ 3c 3

x2

dx

=

?1

2a ( x) 2 dx ?

bx?2 dx +

2

3c x 3 dx

5

= 4a x + b + 9cx3 + C . x5

INTEGRALS 147

3ax

Example 2 Evaluate b2 + c2 x2 dx Solution Let v = b2 + c2x2 , then dv = 2c2 xdx

Therefore,

b2

3ax + c2 x2

dx

=

3a 2c2

dv v

=

3a 2c2

log

b2

+

c2 x2

+C.

Example 3 Verify the following using the concept of integration as an antiderivative.

x3dx

=

x2 x?

+

x3 ? log x + 1 + C

x +1

23

Solution

d x2 dx x ? 2

+ x3 3

?

log

x

+

1

+

C

= 1 ? 2x + 3x2 ? 1 2 3 x+1

1

x3

= 1 ? x + x2 ?

=

x +1

x +1 .

Thus

x

?

x2 2

+

x3 3

? log

x

+1

+

C

=

x3 dx

x +1

Example 4

Evaluate

1+ 1?

x x

dx

,

x

1.

Solution Let I =

1+ x dx = 1? x

1 1 ? x2 dx +

x dx 1 ? x2 = sin?1 x + I1 ,

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