L9

BITS Pilani, K K Birla Goa Campus

BITS PilaniK K Birla Goa Campus

Engineering Mathematics – I

(MATH ZC 161)

Dr. Amit Setia (Assistant Professor)Department of Mathematics

BITS PilaniK K Birla Goa Campus


Chapter 1 & chapter 3Pratap Singh, Review of Elementary Calculus, WILPD- Notes

Summary of lecture 8

• L’Hopital’s Rule

• Rationalization method to find limit

• How to check continuity of a function

• Derivative


• Derivative

• Rules of derivatives

• Higher order derivatives

• Derivative by Chain Rule

• Derivative by Substitution31/08/2012 3Engineering Mathematics – I (MATH ZC 161)

• Integration

• Fundamental Integration Formulae

• Some properties of Integration

• Various methods for integration


• Various methods for integration

31/08/2012 4Engineering Mathematics – I (MATH ZC 161)

Outline of lecture 9

• Integration by parts

• Integration of some standard forms

• Definite Integral

• Properties of definite integral


• Properties of definite integral

• Some examples based on the properties of

definite integral


Don’t forget

You need to

of "Pratap Singh, Review of Elementary Calculus,

remember all the formulae given in Chapter 2


of "Pratap Singh, Review of Elementary Calculus,

WILPD Notes" uploaded on Taxila in a folder

as MATH_ZC161_Lecture_Notes

−


• If u and v are functions of x then the formula

for integration by parts is

duuvdx u vdx vdx dx

= −∫ ∫ ∫ ∫

Article: 4.10

Integration by Parts


u is called I functions and v is called II function.

duuvdx u vdx vdx dx

dx = −

∫ ∫ ∫ ∫


• Rule 1:

If both u and v are integrable functions,

then take that function as the first function

which can be finished by repeated


which can be finished by repeated

differentiation.


Example



• Rule 2:

If the integral contains logarithmic or inverse

trigonometric function,

then take logarithmic or inverse trigometric


then take logarithmic or inverse trigometric

function as the first function and other

function as the II function.

If there is no other function then take 1 as the

second function.


Example





Example

( )

( ) ( ) ( )( )( )

2

2 2

log 1

log 1 log 1

Integrating by parts, we get

III

x dx

dx dx x dx dx

dx = −

∫

∫ ∫ ∫



( ) ( )( )( ) ( ) ( )

( ) ( ) ( )

( ) ( )

2

2

2

2loglog

log 2 log 1

log 2 log

III

dx

xx x x dx

x

x x x dx

x x x x x C

= −

= −

= − − +

∫ ∫ ∫

∫

∫

• Rule 3:

If both functions are integrable and none can

be finished by repeated differentiation, then

any function can be taken as the I function


any function can be taken as the I function

and other as the II function. Repeat the rule of

integration by parts.


3

2

cosec

cosec cosec

Integrating by parts, we get

III

x dx

x x dx=

∫

∫

Example


( ) ( )( )( ) ( )( )

( )

2 2

2

cosec ccosec

cosec

cosec c

c o

ot cosec cot

sec

cot co

osec

cosec co tt

x dx x dx

x x

x dx

x dx

x x x x dx

dx

dx

x x

= −

= −

=

−

−

−

−

−

∫ ∫∫

∫

∫31/08/2012 15Engineering Mathematics – I (MATH ZC 161)

( )( )( )

2

3

3

cosec cot cosec cosec 1

cosec cot cosec cosec

cosec cot cosec cosec

x x x x dx

x x x x dx

x x x dx x dx

= − − −

= − − −

= − − +

∫

∫

∫ ∫


3

3

3

2 cosec cosec cot cosec

2 cosec cosec cot log cosec cot

1cosec cosec co

2

x dx x x x dx

x dx x x x x C

x dx x

⇒ = − +

⇒ = − + − +

⇒ = −

∫ ∫

∫ ∫

∫

∫1

t log cosec cot2 2

Cx x x+ − +


• Rule 4:

That function is taken as the first function

which comes first in the word ILATE

where I = inverse circular function,


where I = inverse circular function,

L = Logarithmic function,

A = Algebraic function,

T = Trigonometric function,

E = Exponential function.


Example



putting sin cosx a dx a dθ θ θ= ⇒ =

Article: 4.11

Some standard forms



putting sin cosx a dx a dθ θ θ= ⇒ =



Please try !

use sec sec tanHint: x a dx aθ θ θ= ⇒ =



2

Please try !

use tanHint: secx a dx a dθ θ θ= ⇒ =

Please try !

do integration by parts,Hint:



2 2

do integration by parts,

take as I function and 1 as II f

H

u

in

nc

t:

tiona x−

Similarly,



The integrals of the above form can be evaluated

Article: 4.12



The integrals of the above form can be evaluated

by writing in any one of the 6 forms just described.

Example



Example





Example





Article: 4.13



To solve the integral of above form, we write

Example

( )1



( )



( )1 becomes

( )2

Putting in first integral 2 5

2&

in 2 integral 1

2

nd

x x t

x dx t

x u

dx du

d

+ =

+ +

=

⇒

⇒ + =

=



dx du⇒ =

( )( )2Evaluate: 3 45x x x dx+ − −∫

Problem



Please try !

Article: 4.14



22

1 2tan

To solve the integral of above fo

sec2 2 2 1

rm, we take

x x dtt dx dt dx

t⇒ ⇒

= = = +



Example



2

2tan

2 1|

x dtt dx

t = =⇒ +

∵



Article: 4.15



( )1 2

To solve the integral of above form, we take

sin cos Denominator Denominatord

a x b x K Kdx + = +

( )

To solve the integral of above form, we take

2sin 3cos 3sin 4cos 3si

2sin 3cos

3sin 4cos

n 4cosd

x x K x x K x

x xd

xx

x

x + = + + +

+

+∫

Example



( )1 22sin 3cos 3sin 4cos 3sin 4cosd

x x K x x K xdx

x + = + + +

1 2

1 2

Equating coefficient of sin and cos , we get

3 4 2

4 3 3

18 1,

x x

K K

K K

K K

∴− =

+ =

⇒ = =



1 2

18 1,

25 25K K⇒ = =

18 1log | 3sin 4cos |

25 25x x x C= + + +

Article: 4.16



( )sin cos Denominator Denominator

where p,q,r

To solve the integral of above form, we

can be evaluated by comparing coefficients of

cos , sin and const

ta

ant terms

ke

da x b x

x

dx

x

C p q r + + = + +

Given integral



The last integral part can be evaluated by using method

described in article 4.14

( ) [ ]( ) ( )

If is a continuous function on ,

Fundamental Theorem of

, and

,

Calculus

f x a b

f x dx F x C= +∫

Article: 4.17 Definite Integral



( ) ( )

( ) ( ) ( )

,

then b

a

f x dx F x C

f x dx F b F a

= +

= −

∫

∫

Example

/22

0 0sin cos cos

I IIx x dx x x x dx

π π = + ∫ ∫





( )To evaluate

if we are taking a substitution ( ),

then find the values of t corresponding to

and .

b

af x dx

x g t

x a x b

=

= =

∫

Remark



and .

If and

when x = a and x = b respectively, then

x a x b

t A t B

= == =

Example





Properties of definite integral







Example



( )of property 7 , as∵

Example



( )of property 4∵



If sin and cos are replaced

by tan and cot or

by cosec and sec respectively

x x

x x

x x

Remark



in the integral I,

then similarly we can evaluate the integral.

Example



( )| of property 3∵



Thanks


Thanks

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