Ex 3.4 (Www.urdulovers123.Blogspot.com)

7/26/2019 Ex 3.4 (Www.urdulovers123.Blogspot.com)

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Edited ByAdnan Shafique Exercise 3.4 (Solutions)

Integration by PartsIf u and v are function of , then

( )uv dx u vdx vdx u dx¢= - ×ò òò òQuestion # 1(i)

Let sin I x x dx= ò Integration by parts

( cos ) ( cos ) (1) I x x x dx= × - - - ×òcos cos x x x dx= - + òcos sin x x x c= - + +

Question # 1(ii)

Let ln I x dx= òln 1 x dx= ×ò

Integrating by parts

1ln I x x x dx

x= × - ×ò

ln x dx= - òln x x c= - +

Question # 1(iii)Let ln I x x dx= ò

Integrating by parts2 2 1

ln2 2

x x I x dx= × - ×ò

21

ln2 2

x x dx= - ò

2 21

ln2 2 2

x xc= - × +

2

1ln2 2

x cæ ö= - +ç ÷è ø

Question # 1(iv)

Let 2 ln I x x dx= ò Do yourself

Question # 1(v)


Question # 1(vi)


Question # 1(vii)

Let 1tan I x dx-= ò1tan 1 x dx-= ×ò


1

2

1tan

1

I x x x dx-= × - ×

+ò

1

2

1 2tan

2 1

x x dx

x

-= -+ò

2

1

2

(1 )1tan

2 1

d xdx x dx

-+

= -+ò

1 21tan ln 1

2 x x c

-= - + +

Question # 1(viii)

Let 2 sin I x x dx= òIntegrating by parts

2 ( cos ) ( cos ) 2 I x x x x dx= - - - ×ò2 cos 2 cos x x x x dx= - + ò

Again integrating by parts

( )2 cos 2 sin sin (1) I x x x x x dx= - + - ò 2 cos 2 sin 2( cos ) x x x x x c= - + - - +

2 cos 2 sin 2cos x x x x x c= - + + +

Question # 1(ix)

Let 2 1tan I x x dx-= òIntegrating by parts

3 31

2

1tan

3 3 1

x x I x dx

x

-= × - ×+ò

3 31

2

1tan

3 3 1

x x x dx

-= -+ò

31

2

1tan

3 3 1

x x x x dx

x

- æ ö= - -ç ÷+è øò

31

2

1 1tan

3 3 3 1

x x x x dx dx

x

-= - + -+ò ò

3 21

2

1 1 1 2tan

3 3 2 3 2 1

x x x x dx

x

-= - × + - ×

+ò2

3 21

2

(1 )1tan

3 6 6 1

d x x x dx x dx-

+= - + -

+ò3 2

1 21tan ln 1

3 6 6

x x x x c-= - + - + +

Question # 1(x)

Let 1tan I x x dx-= òIntegrating by parts

2 2

1

21tan

2 2 1 x x I x dx

x-= - ×

+ò2 2

1

2

1tan

2 2 1

x xdx

x

-= -+ò

2 21

2

1 1 1tan

2 2 1

x xdx

x

- + -= -

+ò2 2

1

2 2

1 1 1tan

2 2 1 1

x xdx

x x

- æ ö+= - -ç ÷+ +è ø

ò2

1

2

1 1tan 1

2 2 1

xdx

x

- æ ö= - -ç ÷+è øò

u x=sinv x=

lnu xv x

==

2

lnu x

v x

==

3lnu x

v x

==

4

lnu x

v x

==

1tan1

u xv

-==

2u x=

sinv x=

cosu xv x==

1

2

tanu x

v x

-==

1tanu xv x

-==

ln1

u xv

==

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FSc-II Ex- 3.4 -

21

2

1 1 1tan

2 2 2 1

xdx dx

x

-= - ++ò ò

21 11 1

tan tan2 2 2

x x x c- -= - + + Ans.

Question # 1(xi)

Let 3 1tan I x x dx-=

ò Integrating by parts4 4

1

2

1tan

4 4 1

x x I x dx-= × - ×

+ò

4 4

1

2

1tan

4 4 1

x xdx

x

-= - -+ò

41

2

2

tan4

1 11

4 1

x x

x dx x

-=

æ ö- - +ç ÷+è øò

4

1 2

2

1 1 1 1tan

4 4 4 4 1

x x dx dx dx

x

-= - + -+ò ò ò

4 31 11 1 1

tan tan4 4 3 4 4

x x x x c- -= - + - +

4 31 11 1

tan tan4 12 4 4

x x x x c- -= - + - +

Question # 1(xii)

Let 3 cos I x x dx= ò

Do yourself as Question # 1(viii). Integrate by parts three times.

Question # 1(xiii)1sin I x dx-= ò1sin 1 x dx-= ×ò


1

2

1sin

1 I x x x dx

x

-= × - ×-

ò1

1 2 2

sin (1 ) ( ) x x x x dx

--

= - -ò 11 2 21

sin (1 ) ( 2 )2

x x x x dx--= + - -ò

11 2 221

sin (1 ) (1 )2

d x x x x dx

dx

--= + - -ò1

12 21 (1 )1

sin12

12

x x x c

- +

- -= + +

- +

12 2

1 (1 )1

sin 122

x

x x c

- -= + +

1 2sin 1 x x x c-= + - +

Question # 1(xiv)

Let 1sin I x x dx-= ò Integrating by parts

2 21

2

1sin

2 2 1

x x I x dx

x

-= × - ×-

ò

2 21

2

1sin2 2 1

x x x dx

x

- -

= + -ò

2 21

2

1 1 1sin

2 2 1

x x x dx

x

- - -= +

-ò

2 21

2 2

1 1 1sin

2 2 1 1

x x x dx

x x

- æ ö-= + -ç ÷

- -è øò

21 2

2

1 1

sin 12 2 1

x

x x dx x

- æ ö

= + - -ç ÷-è øò

21 2

2

1 1 1sin 1

2 2 2 1

x x x dx dx

x

-= + - --

ò ò2

1 1

1

1 1sin sin

2 2 2

x I x I x- -Þ = + - ….. (i)

Where 21 1 I x dx= -ò

Put sin x q = cosdx d q q Þ =

21 1 sin cos I d q q q Þ = -ò

2cos cos d q q q = ò

2cos d q q = ò 1 cos 2

2 d

q q

+æ ö= ç ÷è øò

( )1

1 cos 22

d q q = +ò

1 sin 2

2 2 c

q q

é ù= + +ê úë û

1 2sin cos

2 2 c

q q q

é ù= + +ê úë û

21sin 1 sin

2 cq q q é ù= + - +ê úë û

1 21sin 1

2 x x c-é ù= + - +ê úë û

Using value of 1 I in ( )i

( )2

1 1 2

1

1 1sin sin 1

2 2 2

1sin

2

x I x x x x c- -

-

é ù= + + - +ê úë û

-

2

1 1 2

1

1 1 1sin sin 12 4 4 2

1sin

2

x x x x x c

x

- -

-

= + + - +

-

21 1 21 1 1

sin sin 12 4 4 2

x I x x x x c- -Þ = - + - +

Question # 1(xv)

Let sin cos x I e x x dx= ò1

2 sin cos2

xe x x dx= ×ò

1sin 2

2 x

e x dx= ò sin 2 2 sin cos x x x=Q


1 cos 2 cos 2

2 2 2 x x x x

I e e dx- -é ù= × - ×ê úë ûò

1 1cos 2 cos 2

4 4 x x

e x e x dx= - + òAgain integrating by parts

1 1 sin 2 sin 2cos2

4 4 2 2 x x x x x

I e x e eæ ö= - + × -ç ÷è øò

1sinu x-=v x=

xu e=

sin2v x=

2

2 4

4 2

2

2

1

1

1

1

x

x x

x x

x

x

- -

+ +

-

+

+

-

- -

3

cosu xv x

==

1

3

tanu x

v x

-=

=

1sin1

u xv

-==





FSc-II Ex- 3.4 -

1 1 sin 2 1cos 2 sin 2

4 4 2 2 x x x x

e x e e xæ ö= - + × -ç ÷è øò

1 1 sin 2

cos24 4 2

x x xe x e I c

æ ö= - + × - +ç ÷è ø

1 1 1

cos 2 sin 24 8 4

x xe x e x I c= - + - +

1 1 1cos 2 sin 2

4 4 8 x x I I e x e x cÞ + = - + +

5 1 1cos 2 sin 2

4 4 8 x x I e x e x cÞ = - + +

1 1 4cos 2 sin 2

5 10 5 x x I e x e x cÞ = - + +

Question # 1(xvi)

Let sin cos I x x x dx= ò1

2 sin cos2 x x dx= ×

ò1sin2

2 x dx= ×ò


1 cos 2 cos 2(1)

2 2 2

x x I x dx

é - - ùæ ö æ ö= -ç ÷ ç ÷ê úè ø è øë ûò

Now do yourself

Question # 1(xvii)

Let 2cos I x x dx= ò1 cos 2

2 x dx+æ ö= ç ÷

è øò

( )1

1 cos 22

x x dx= +ò1 1

cos22 2

dx x xdx= +ò ò

21 1 sin 2 sin 2

(1)2 2 2 2 2

x x xdx

é ù= × + - ×ê úë ûò2

1 1sin 2 sin 2

4 4 4

x x x dx= + × - ò

2 1 1 cos 2

sin24 4 4 2

x x x c

-æ ö= + × - +ç ÷è ø

2 1 1sin 2 cos 2

4 4 8

x x x c= + × + +

Question # 1(xviii)

Let 2sin I x x dx= ò1 cos 2

2

x x dx

-æ ö= ç ÷è øò ( )

11 cos 2

2 x x dx= -ò

1 1 cos22 2

x dx x x dx= -ò ò

Integrating by parts21 1 sin 2 sin 2

.(1)2 2 2 2 2

x x x I x dx

é ù= - × -ê úë ûò2 1 1

sin 2 sin 24 4 4

x x x dx= - + ò

2 1 1 cos 2

sin 24 4 4 2

x x x c

-æ ö= - + +ç ÷è ø

2 1 1

sin 2 cos 24 4 8

x

x x c= - - +

Question # 1(xix)

Let 2(ln ) I x dx= ò( )

2ln 1 x dx= ×ò


( ) ( )2 1

ln 2 ln I x x x x dx

x

= × - × ×

ò( ) ( )2

ln 2 ln x x x dx= - ò Again integrating by parts

( )2 1

ln 2 ln I x x x x x dx x

é ù= - × - ×ê úë ûò

( )2

ln 2 ln 2 x x x x dx= - + ò( )

2ln 2 ln 2 x x x x x c= - + +

Question # 1(xx)

Let 2ln(tan ) sec I x x dx= òIntegrating by parts

21ln(tan ) tan tan sec

tan I x x x x dx

x= × - × ×ò

( ) 2tan ln tan sec x x x dx= - ò( )tan ln tan tan x x x c= - +

Question # 1(xxi)

Let1

2

sin

1

x x I dx

x

-

=

-

ò

1

2

1sin ( )

1 x x dx

x

-= ×-

ò

( )1

1 2 21sin 1 ( 2 )

2 x x x dx

--= - × - -òIntegrating by parts

112 2

1

1

12 2

2

(1 )1sin

121

2

(1 ) 1

1 112

x I x

xdx

x

- +

-

- +

é-ê

= - ×ê- +êë

ù- ú- × ú

-- + úû

ò

12 2

1

12 2

2

(1 )1sin

12

2

(1 ) 1

1 12

x x

xdx

x

-

é-ê

= - ×êêë

ù- ú

- × ú- úû

ò

12 12

12(1 ) sin 2

2 x dx-é ù

= - - -ê úë û

ò2 11 sin x x dx-= - - + ò

2 11 sin x x x c-= - - + +

2 11 sin x x x c-= - - +

Question # 2(i)

Let 4tan I xdx= ò2 2tan tan x x dx= ×ò

sin2u xv x

==

cos2u xv x

==

cos2u xv x==

( )2

ln1

u xv

==

2ln(tan )secu xv x

==

( )

1

22

1

1 ( 2 )

sin

x x

u x

v -

-

- -

=

=





FSc-II Ex- 3.4 -

( )2 2tan sec 1 x dx= -ò( )2 2 2tan sec tan x x dx= -ò

2 2 2tan sec tan x dx x dx= -ò ò2 2tan (tan ) (sec 1)

d x dx x dx

dx= - -ò ò

2 12tan sec

2 1 x x dx dx

+= - -

+ ò ò31

tan tan3

x x c= - - +

Question # 2(ii)

Let 4sec I x dx= ò2 2(sec ) (sec ) x dx= ×ò

2 2(1 tan ) (sec ) x dx= + ×ò2 2 2sec tan secdx x x dx= +ò ò

2tan (tan ) (tan )d

x x dxdx

= + ò

3tan

tan3

xc= + +

Question # 2(iii)

Let sin2 cos x I e x x dx= ò

( )1

2 sin 2 cos2

xe x x dx= ò

( )1sin(2 ) sin(2 )

2 xe x x x x dx= + + -ò

( )1

sin 3 sin2

xe x x dx= +ò1 1

sin 3 sin2 2

x xe x dx e x dx= +ò ò

1 2

1 1

2 2 I I = + ………. (i)

Where 1 sin3 x I e x dx= ò and 2 sin x I e x dx= òSolve 1 I and 2 I as in Q # 1(xv) and put value of

1 I and 2 I in ( )i .

Question # 2(iv)3tan sec I x x dx= ×ò

2tan tan sec x x dx= × ×ò( )2sec 1 sec tan x x dx= - ×ò

Put sect x= sec tandt x x dxÞ =

So ( )2 1 I t dt = -ò3

3t t c= - +

3sec

sec3

xc= - +

Question # 2(v)

Let 3 5 x I x e dx= òIntegrating by parts

5 53 23

5 5

x xe e I x x dx= × - ×ò

3 5 2 51 3

5 5 x x x e x e dx= - ò

Again integrating by parts5 5

3 5 21 32

5 5 5 5

x x x e e

I x e x x dxé ù

= - × - ×ê úë û

ò

3 5 2 5 51 3 6

5 25 25

x x x x e x e x e dx= - +

òAgain integrating by parts

3 5 2 5

5 5

1 3

5 256

(1)25 5 5

x x

x x

I x e x e

e edx

= -

é ù+ × - ×ê ú

ë ûò

3 5 2 5 5 51 3 6 6

5 25 125 125 x x x x x e x e xe e dx= - + - ò

53 5 2 5 51 3 6 6

5 25 125 125 5

x x x x e

x e x e xe c= - + - × +

53 23 6 6

5 5 25 125

x

e x x x cæ ö= - + - +ç ÷è ø

Question 2(vi)

Let sin 2 x I e x dx-= òIntegrating by parts

cos 2 cos 2( 1)

2 2 x x x x

I e e dx- -- -= × - × -ò

1 1cos 2 cos 2

2 2 x xe x e x dx- -= - - ò

Again integrating by parts1 1 sin 2

cos22 2 2

sin2( 1)

2

x x

x

x I e x e

xe dx

- -

-

é= - - ×êëù- × - úûò

1 1 1

cos 2 sin 2 sin 22 4 4

x x xe x e x e x dx- - -= - - - ò

1 1 1

cos 2 sin 22 4 4

x x I e x e x I c- -Þ = - - - +

1 1 1cos 2 sin 2

4 2 4 x x I I e x e x c- -Þ + = - - +

5 1 1cos 2 sin 2

4 2 4 x x I e x e x c- -Þ = - - +

2 1 4

cos 2 sin 25 5 5

x x I e x e x c- -Þ = - - +

( )1 4

2cos 2 sin 25 5

xe x x c-= - + +

Question # 2(vii)

Let 2 cos3 x I e x dx= ×ò Do yourself as above

Question # 2(viii)3cosec I x dx= ò

2cosec cosec x dx= ×òIntegrating by parts

( ) ( )( )csc cot cot csc cot I x x i x x x dx= - - -ò2cosec cot cosec cot x x x x dx= - - ò

( )2cosec cot cosec cosec 1 x x x x dx= - - -ò

( )3cosec cot cosec cosec x x x x dx= - - -ò

3

x

u x

v e

==

2

x

u x

v e

==

sin2

xu ev x

-==

2cosec

cosec

u x

v x

==





FSc-II Ex- 3.4 -

3cosec cot cosec cosec x x x dx x dx= - - +ò òcosec cot ln cosec cot x I x x c= - - + - +

cosec cot ln cosec cot I I x x x x cÞ + = - + - +

2 cosec cot ln cosec cot I x x x x cÞ = - + - +

1 1 1

csc cot ln csc cot2 2 2 I x x x x cÞ = - + - +

Question # 3

Let sinax I e bx dx= òIntegrating by parts

cos cos( )

ax axbx bx I e e a dx

b b

æ ö æ ö= - - - ×ç ÷ ç ÷è ø è øò

coscos

axaxe bx a

e bx dxb b

= - + òAgain integrating by parts

cos sin sinaxax axe bx a bx bx

I e e a dxb b b b

é ù= - + - ×ê úë ûò

2

2 2

cossin sin

axax axe bx a a

e bx e bx dxb b b

= - + - ò

2

12 2

cossin

axaxe bx a a

e bx I cb b b

= - + - +

2

12 2

cossin

axaxa e bx a

I I e bx cbb b

Þ + = - + +

( )

2 2

12 2cos sin

axb a e

I b bx a bx cb b

æ ö+Þ = - + +

ç ÷è ø

( ) ( )2 2 2

1sin cosax

b a I e a bx b bx b cÞ + = - +

Put cosa r q = & sinb r q =

Squaring and adding

( )2 2 2 2 2cos sina b r q q + = +

2 2 2 (1)a b r Þ + = 2 2r a bÞ = +

Also

sin

cos

b r

a r

q

q = tan

b

a q Þ =

1tan b

aq

-Þ =

So

( ) )(2 2

2

1

cos sin sin cosaxb a I e r bx r bx

b c

q q + = -

+

( ) )(2 2

2

1

sin cos cos sinaxb a I e r bx bx

b c

q q + = -

+

( ) ( )2 2 2

1sinax

a b I e r bx b cq Þ + = - +

Putting value of r and q

( )2 2 2 2 1 2

1sin tanax b

a b I e a b bx b ca

-æ ö+ = + - +ç ÷è ø

2 2 21

12 2 2 2sin tan

( )

axa b b b I e bx c

aa b a b

-+ æ öÞ = - +ç ÷+ +è ø

1

2 2

1sin tan

ax b I e bx c

aa b

-æ öÞ = - +ç ÷è ø+

Where2

12 2

bc c

a b=

+

Question # 4(i)

Let 2 2 I a x dx= -ò2 2 1a x dx= - ×ò


( ) ( )1

2 2 2 2 212

2 I a x x x a x x dx

-= - × - × - × -

ò

( )

22 2

12 2 2

xa x dx

a x

-= - -

-ò

( )

2 2 22 2

12 2 2

a x aa x dx

a x

- -= - -

-ò

( ) ( )

2 2 22 2

1 12 2 2 22 2

a x a x a x dx

a x a x

æ öç ÷-

= - - -ç ÷ç ÷- -è ø

ò

2

2 2 2 2

2 2

aa x a x dx dx

a x= - - - +

-ò ò

2 2 2

2 2

1 I x a x I a dx

a xÞ = - - +

-ò

2 2 2 1 x I I x a x a Sin ca

-Þ + = - + +

2 2 2 12 x

I x a x a Sin ca

-Þ = - + +

2 2 2 11 1 1

2 2 2

x I x a x a Sin ca

-

Þ = - + +

Review

2 2

2 2ln

dx x x a c

x a· = + - +

-ò

2 2

2 2ln

dx x a c

x a· = + + +

+ò

Question # 4(ii)

Let 2 2 I x a dx= -ò2 2 1a dx= - ×ò


( ) ( )1

2 2 2 2 212

2 I x a x x x a x dx

-= - × - × - ×ò

( )

22 2

12 2 2

x x x a dx

x a

= - -

-ò

( )

2 2 22 2

1

2 2 2

x a a x x a dx

x a

- += - -

-

ò

( ) ( )

2 2 22 2

1 12 2 2 22 2

x a a x x a dx

x a x a

æ öç ÷-

= - - +ç ÷ç ÷- -è ø

ò

2

2 2 2 2

2 2

a x x a x a dx dx

x a= - - - -

-ò ò

2 2 2

2 2

1 I x x a I a dx

x aÞ = - - -

-ò

2 2

1u a xv

= -=

sin

axu ev bx

==

2 2

1u x av

= -

=





FSc-II Ex- 3.4 -

2 2 2 2 2ln I I x x a a x x a cÞ + = - - + - +

2 2

2 2ln

dx x x a c

x a= + - +

-òQ

2 2 2 2 22 ln I x x a a x x a cÞ = - - + - +

22 2 2 21 1ln

2 2 2a I x x a x x a cÞ = - - + - +

Question # 4(iii)

Let 24 5 I x dx= -ò24 5 1 dx= - ×ò

Integrating by parts1

2 2 214 5 (4 5 ) ( 10 )

2 I x x x x x dx

-= - × - × - × -ò

2

2254 5

(4 5 ) x x dx x-= - × - -ò

22

2

4 5 44 5

(4 5 )

x x dx

x

- -= - × -

-ò

22

1 12 22 2

4 5 44 5

(4 5 ) (4 5 )

x x x dx

x x

æ ö-ç ÷= - × - -

ç ÷ç ÷- -è øò

12 2 2

12

2

44 5 (4 5 )

(4 5 )

x x dx

x

æ öç ÷= - × - - -ç ÷ç ÷

-è ø

ò

2 2

2

14 5 4 5 4

4 5 x x x dx dx

x= - × - - +

-ò ò

2

2

14 5 4

45

5

I x x I dx

x

Þ = - × - +æ ö-ç ÷è ø

ò

2

2

14 5 4

45

5

I I x x dx

x

Þ + = - × +

-ò

2

2

2

4 12 4 55 2

5

I x x dx

x

Þ = - × +æ ö

-ç ÷è ø

ò

2 1

1

44 5

5 2 5

x x x Sin c

- æ ö= - × + +ç ÷

è ø

1

2 2

dx xSin

aa x

-=-

òQ

2 1

1

54 14 5

2 2 22 5

x x I x Sin c- æ ö

Þ = - + +ç ÷

è ø

2 1 524 5

2 25

x x x Sin c

- æ ö= - + +ç ÷

è ø

Where 1

1

2c c=

Question # 4(iv)

Let 23 4 I x dx= -òSame as above.

Question # 4(v)Same as Q # 4(ii)

Use 2

2ln 4

4

dx x c

x= + + +

+ò

Question # 4(vi)

Let

2 ax

I x e dx= ò Do yourself as Question # 2(v)

Important Formula

Since ( )( ) ( ) ( )ax ax axd d d e f x e f x f x e

dx dx dx= +

( ) ( ) ( )ax axe f x f x e a¢= + ×

[ ]( ) ( )axe a f x f x¢= +

On integrating

( ) [ ]( ) ( ) ( )ax axd

e f x dx e a f x f x dxdx

¢= +

ò ò[ ]( ) ( ) ( )ax axe f x e a f x f x dx¢Þ = +ò [ ]( ) ( ) ( )ax axe a f x f x dx e f x c¢Þ + = +ò

Question # 5(i)

Let1

ln x I e x dx x

æ ö= +ç ÷

è øò1

ln xe x dx x

æ ö= +ç ÷è øò

Put ( ) ln f x x= 1

( ) f x¢Þ =

So ( )( ) ( ) x I e f x f x dx¢= +ò ( ) xe f x c= + ln xe x c= +

Question # 5(ii)

Let ( )cos sin x I e x x dx= +ò( )sin cos xe x x dx= +ò

Put ( ) sin f x x= ( ) cos f x x¢Þ =

So ( )( ) ( ) x I e f x f x dx¢= +ò ( ) xe f x c= +

sin xe x c= +

Question # 5(iii)

Let 1

2

1sec

1

ax I e a x dx x x

-é ù= +ê ú

-ë ûò

Put 1( ) sec f x x-= 2

1( )

1

f x

x x

¢Þ =

-So [ ]( ) ( )ax I e a f x f x dx¢= +ò

( )axe f x c= + 1secaxe x c-= +

Question # 5(iv)

Let 3

2

3sin cos

sin

x x x I e dx

x

-æ ö= ç ÷è øò

3

2 2

3sin cos

sin sin

x x xe dx

x x

æ ö= -ç ÷

è ø

ò





FSc-II Ex- 3.4 -

3 1 cos3

sin sin sin x x

e dx x x x

æ ö= -ç ÷×è øò

( )3 3csc csc cot xe x x x dx= -òPut ( ) csc f x x= ( ) csc cot f x x x¢Þ = -

( )3 3 ( ) ( ) x I e f x f x dx¢Þ = +ò 3 ( ) xe f x c= + 3 csc xe x c= +

Question 5(v)

Let ( )2 sin 2cos x I e x x dx= - +ò *Correction

( )2 2cos sin xe x x dx= -òPut ( ) cos f x x= ( ) sin f x x¢Þ = -

So ( )2 2 ( ) ( ) x I e f x f x dx¢= +ò 2 ( ) xe f x c= +

2 cos xe x c= +

Question # 5(vi)

Let( )

21

x xe

I dx x

=+

ò

2

(1 1)

(1 )

x x edx

x

+ -=

+ò

2 2

1 1

(1 ) (1 )

x xe dx

x x

é ù+= -ê ú+ +

ë û

ò

2

1 1

(1 ) (1 )

xe dx x x

é ù= -ê ú+ +ë û

ò

Put 11( ) (1 )

1 f x x

x-= = +

+

2

2

1( ) (1 )

(1 ) f x x

-¢Þ = - + = -+


1

1 xe c

xæ ö= +ç ÷+è ø

Question # 5(vii)

Let ( )cos sin x I e x x dx-= -ò( )( 1)sin cos xe x x dx-= - +ò

Put ( ) sin f x x= '( ) cos f x xÞ =

So ( )( 1) ( ) ( ) x I e f x f x dx- ¢= - +ò ( ) xe f x c-= +

sin xe x c-= +

Question # 5(viii)

Let

1

2

tan

1

m xe

I dx x

-

=+ò

1

2

tan 1

1

m xe dx-

= ×+ò

Put 1tant x-= 2

1

1dt dx

xÞ =

+

Somt

I e dt = ò

mt ec

m= +

1tan1 m xe cm

-= +

Important Integral

Let tan I x dx= òsin

cos

xdx

x= ò

Put cost x= sindt x dxÞ = -

sindt x dxÞ - =

Sodt

I t

-= ò

dt

t = -ò

ln t c= - +

ln cos x c= - +

1ln cos x c-= + ln ln mm x x=Q

1

lncos

c x

= + ln sec x c= +

tan ln sec x dx x cÞ = +òSimilarly, we have

cot ln sin x dx x c= +ò

Question # 5(ix)

Let 21 sin

x I dx x

=-ò

2 1 sin

1 sin 1 sin

xdx

x

+= ×

- +ò

2

2 (1 sin )

1 sin

x xdx

x

+=

-ò

2

2 2 sin )

cos

x x xdx

x

+= ò

2 2

2 2 sin

cos cos

x x xdx

x x

æ ö= +ç ÷

è øò

2

2 2 sincos coscos

x x xdx dx x x x

= +×ò ò

22 sec 2 sec tan x x dx x x x dx= +ò òIntegrating by parts

2 tan tan 1

2 sec sec (1)

I x x x dx

x x dx

é ù= × - ×ë ûé ù+ × -ë û

òò

2 tan ln sec

2 sec ln sec tan

x x x

x x x c

= é × - ùë û+ é × - + ù +ë û

2 tan 2ln sec2 sec 2ln sec tan

x x x x x x c

= -+ - + +

Question # 5(x)

Let2

(1 )

(2 )

xe x I dx

x

+=

+ò

2

(2 1)

(2 )

xe xdx

x

+ -=

+ò

2 2

2 1

(2 ) (2 )

x xe dx

x x

æ ö+= -ç ÷

+ +è ø

ò





FSc-II Ex- 3.4 -

( )1 2(2 ) (2 ) xe x x dx- -= + - +òPut 1( ) (2 ) f x x -= + 2( ) (2 ) f x x -¢Þ = - +


1(2 ) xe x c-= + +

2

xe

c x

= ++

Question # 15(xi)

Let1 sin

1 cos x x

I e dx x

-æ ö= ç ÷-è øò

2

1 2sin cos2 2

2sin2

x

x x

e dx x

æ ö-ç ÷= ç ÷

ç ÷è ø

ò

2 2

2sin cos1 2 2

2sin 2sin2 2

x

x x

e dx x x

æ öç ÷

= -ç ÷ç ÷è ø

ò

21cosec cot

2 2 2 x x x

e dxæ ö= -ç ÷è øò

21cot cosec

2 2 2 x x x

e dxæ ö= - +ç ÷è øò

Put ( ) cot2

f x = - 2 1( ) cosec

2 2

x f x¢Þ = ×

21( ) cosec

2 2 f x¢Þ =

So ( )( ) ( ) x I e f x f x¢= +ò ( ) xe f x c= +

cot2

x xe c

æ ö= - +ç ÷è ø

cot2

x xe c= - + .



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