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Exercise 3.2 (Solutions) Calculus and Analytic Geometry, MATHEMATICS 12

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Theorem on Anti-Derivatives i) ( ) ( )cf x dx c f x dx= where c is constant. ii) [ ]( ) ( ) ( ) ( )f x g x dx f x dx g x dx =

Important Integral

Since ( )1 1n nd x n xdx+ = +

Taking integral w.r.t x

( )1 1n nd x dx n x dxdx+ = +

( )1 1n nx n x dx+ = +

1

1

nn xx dx n

+

=+ where 1n -

If 1n = - then 1 1x dx dxx

- =

Since 1lnd xdx x=

Therefore 1 lndx x cx = + Note: Since log of negative numbers does not exist therefore in above formula mod assure that we are taking a log of +ive quantity.

Question # 1(i) ( )23 2 1x x dx- + 23 2x dx xdx dx= - +

2 1 1 1

3 22 1 1 1x x x c

+ +

= - + ++ +

3 2

3 23 2x x x c= - + +

3 2x x x c= - + +

Question # 1(ii)

1x dxx

+

1 12 2x x dx

- = +

1 12 2x dx x dx

-= +

1 11 12 2

1 11 12 2

x x c+ - +

= + ++ - +

3 12 2

3 12 2

x x c= + +

3 12 22 23 x x c= + + Ans.

Question # 1(iii)

( )1x x dx+12 1x x dx = +

32x x dx = +

32x dx x dx= +

3 1 1 12

3 1 112

x x c+ +

= + +++

5

22

5 22

x x c= + +

5222 1

5 2x x c= + +

Important Integral

Since ( ) ( )1 ( 1)n nd ax b n ax b adx++ = + +

Taking integral

( ) ( )1 ( 1)n nd ax b dx n ax b a dxdx++ = + +

( ) ( )1 ( 1)n nax b n a ax b dx+ + = + +

( ) ( )1

( 1)

nn ax bax b dx n a

++ + =

+

Question # 1(iv)

( )122 3x dx+

( )1 122 3

1 1 22

xc

++= +

+

( )322 3

3 22

xc

+= +

( )321 2 33 x c= + +

Question # 1(v)

( )21x dx+ ( )2( ) 2 1x x dx= + + ( )

122 1x x dx = + +

( )122x dx x dx dx= + +

1 11 1 2

21 1 1 12

x x x c++

= + + ++ +

3

2 222 3

2

x x x c= + + +

3

2 242 3x x x c= + + +

Question # 1(vi)

2

1x dxx

-

1 2x dxx = + -

1 2xdx dx dxx= + -

2

ln 22x x x c= + - +

FSc-II / Ex- 3.2 - 2

Question # 1(vii)

3 2x dxx+

1 23 2x dx

x+=

1 2 1 23 2x dxx x

= + ( )1 2 1 23 2x x dx-= + 1 2 1 23 2x dx x dx-= +

Now do yourself.

Question # 1(viii)

( )1y y dyy+

( )( )2

1y ydy

y

+=

( )1y dyy

+=

1y dyy y

= +

1 12 2y y dy

- = +

1 12 2y dy y dy

-= +

1 11 12 2

1 11 12 2

y y c+ - +

= + ++ - +

3 12 2

3 12 2

y y c= + +

3 12 22 23 y y c= + +

Question # 1(ix)

( )21

dq

qq

-

( )2 1d

q qq

q

- +=

2 1 dq q qq q q

= - +

1 12 22 dq q q = - +

No do yourself

Question # 1(x) Do yourself as above

Important Integral

We know ax axd e a edx =

Taking integral ax axd e dx a e dxdx =

ax axe a e dx = ax

ax ee dx a =

Also note that ( )

( )ax b

ax b ee dx a

++ =

Question # 1(xi)

2x x

xe e dx

e+

2x x

x xe e dxe e

= +

( )1xe dx= +

xe dx dx= + xe x c= + + Ans

Question # 2(i)

dxx a x b+ + +

dx x a x bx a x b x a x b

+ - += + + + + - +

x a x b dxx a x b+ - +=+ - -

( ) ( )1 12 2x a x b

dxa b+ - +

=-

( ) ( )1 12 21 x a dx x b dxa b

= + - + -

( ) ( )1 112 21

1 11 12 2

x a x bca b

+ + + = - + - + +

( ) ( )3 32 21

3 32 2

x a x bca b

+ + = - + -

( ) ( ) ( )3 32 22

3x a x b c

a b = + - + + -

Ans.

Important Integral

Since 1 21

1d Tan xdx x

- =+

Also ( )1 211d Cot xdx x

-- =+

Therefore 121

1dx Tan x

x-=

+ or 1Cot x--

Similarly 12

11

dx Sin xx

-=-

or 1Cos x--

12

11

dx Sec xx x

-=-

or 1Csc x--

Question # 2(ii)

2

211

x dxx

-+

221

1dx

x = - + +

212

1dx dx

x= - +

+ 12x Tan x c-= - + +

Question # 2(iii) dx

x a x+ +

dx x a xx a x x a x

+ -= + + + -

x a x dxx a x+ -=+ -

( ) ( )1 12 2x a x

dxa+ -

=

2 2

2

11 1

1

2

x x

x+ +

-+ -

- -

FSc-II / Ex- 3.2 - 3

( ) ( )1 12 21 x a dx x dxa

= + -

( ) ( )1 112 21

1 11 12 2

x a xca

+ + = - + + +

( ) ( )3 32 21

3 32 2

x a xca

+ = - +

( )33222

3 x a x ca

= + - +

Ans.

Question # 2(iv)

( )322a x dx-

( )( )

3 1223 1 22

a xc

+-= +

+ -

( )( )

522

5 22

a xc

-= +

-

( )522

5a x

c-

= - + Question # 2(v)

( )31 x

x

edx

e+

( )2 31 3 3x x x

x

e e edx

e+ + +

=

2 31 3 3x x x

x x x xe e e dx

e e e e

= + + +

( )23 3x x xe e e dx-= + + +

2

3 31 2

x xxe ex e c

-

= + + + +-

213 3 2x x xe x e e c-= - + + + +

Important Integrals

We know cos sind ax a axdx = -

Taking integral

cos sind ax dx a ax dxdx = - cos sinax a ax dx = -

cossin axax dx a = -

Also sin cosd ax a axdx =

sincos axax dx a\ = Similarly

2 tansec axax dx a=

2 cotcosec axax dx a= -

secsec tan axax ax dx a= csccsc cot axax ax dx a= -

Also note that

( ) ( )cossin ax bax b dx a+

+ = -

( ) ( )sincos ax bax b dx a+

+ = and so on.

Question # 2(vi)

cos( )sin( ) a b xa b x dx a b++ = -

+

Question # 2(vii) 1 cos 2x dx- 22sin x dx= 2

1 cos 2sin 2xx -=Q

2 sin x dx= ( )2 cos x c= - + 2 cos x c= - +

Important Formula

Q [ ] ( )[ ]1( ) 1 ( ) ( )n nd df x n f x f xdx dx+ = +

[ ] ( )[ ]1( ) 1 ( ) ( )n nd f x n f x f xdx+ = +

Taking integral

[ ] ( )[ ]1( ) 1 ( ) ( )n nd f x dx n f x f x dxdx+ = +

[ ] ( ) [ ]1( ) 1 ( ) ( )n nf x n f x f x dx+ = +

[ ] [ ]( )

1( )( ) ( )

1

nn f xf x f x dx

n

+

=+ ; 1n -

Also 1ln ( ) ( )( )d f x f xdx f x

=

Taking integral ( )ln ( ) ( )

f xf x dxf x

=

i.e. ( ) ln ( )( )f x dx f x cf x

= +

Question # 2(viii)

Let 1lnI x dxx=

Put ( ) lnf x x= 1( )f x x =

So [ ]( ) ( )I f x f x dx=

[ ]1 1( )

1 1f x

c+

= ++

[ ]2( )

2f x

c= +

( )2ln

2x

c= +

FSc-II / Ex- 3.2 - 4

Question # 2(ix)

2sin x dx 1 cos 2

2x dx- =

1 1 cos 22 2 x dx = -

1 1 cos 22 2dx x dx= -

1 1 sin 22 2 2xx c= - +

1 1 sin 22 4x x c= - +

Question # 3(x)

11 cos dxx+

2

1

2cos 2

dxx= 2 1 coscos 2 2

x x+=Q

21 sec2 2x dx=

tan1 22 1

2

xc= + tan 2

x c= +

Alternative

11 cos dxx+ 1 1 cos

1 cos 1 cosx dxx x

-= + -

21 cos1 cos

x dxx

-=-

21 cos

sinx dx

x-=

2 21 cos

sin sinx dx

x x = -

2 coscosec sin sinxx dxx x

= - 2cosec cosec cotxdx x x dx= -

( )cot cosecx x c= - - - + cosec cotx x c= - +

Question # 2(xi)

Let 2ax bI dx

ax bx c+=

+ + Put 2( ) 2f x ax bx c= + + ( ) 2 2f x ax b = +

( )( ) 2f x ax b = + 1 ( )2 f x ax b = +

So 12 ( )

( )f x

I dxf x

=

( )12 ( )f x dxf x

= 11 ln ( )2 f x c= +

2 11 ln2 ax bx c c= + + +

Review ( ) ( )2sin cos sin sina b a b a b = + + - ( ) ( )2cos sin sin sina b a b a b = + - - ( ) ( )2cos cos cos cosa b a b a b = + + - ( ) ( )2sin sin cos cosa b a b a b - = + - -

Question # 2(xii) cos3 sin 2x x dx 1 2cos3 sin 22 x x dx=

[ ]1 sin(3 2 ) sin(3 2 )2 x x x x dx= + - -

[ ]1 sin 5 sin2 x x dx= -

( )1 cos5 cos2 5x x c = - - - +

1 cos5 cos2 5x x c = - - +

Question # 2(xiii)

cos 2 11 cos 2x dxx

-+

1 cos 21 cos 2x dxx

-= -+

2

22sin2cos

x dxx

= - 2tan x dx= - ( )2sec 1x dx= - - 2sec x dx dx= - + tan x x c= - + +

Question # 2(xiv) 2tan x dx ( )2sec 1x dx= -

2sec x dx dx= - tan x x c= - +

Important Integral

Since ( )1lnd dax b ax bdx ax b dx+ = ++

1lnd ax b adx ax b + = +

On Integrating

1ln ax b a dxax b + = +

ln1 ax bdxax b a

+ =

+

The End

Thursday, 11 November 2012 By mathcity.org, mathcity@gmail.com

Found Error

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Error Analysts 1- Muhammad Hussnain Khadim - (20011-13)

PISH AL-HASA KSA

2

2

1 cos 2sin 21 cos 2cos 2

xxxx

-=

+=

Q