Ex_3_2_FSC_part2
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Exercise 3.2 (Solutions) Calculus and Analytic Geometry, MATHEMATICS 12
Available online @ http://www.mathcity.org, Version: 1.0.4
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, [email protected]
Found Error
Tell us at
http://www.mathcity.org/error
Error Analysts 1- Muhammad Hussnain Khadim - (20011-13)
PISH AL-HASA KSA
2
2
1 cos 2sin 21 cos 2cos 2
xxxx
-=
+=
Q