IIII S

11 / / /
111 -

Integration
 ·
Ja
=

/132 1)" +

= "(B
:
=(
: +
: 5 -

1 56
=
.

=
5

3
= 2
Gradient of normal : -I
Equation :

y 3 -
=
-

2(x 3) -

y -
3 = Ex 3 -
+

y :
-

[x + 2
[x + y =
z
9
x +
2y =

i
J y 3 - 3 C +
=

3 = 33)3 + C -

3 : 9 + C

Just c

y
=

:
-
6

35 -- -
6
2
J6(42-3)
=

·
 Ezei
i)y iii) Va
= gynda 2
2x x +
y
= + =

:
2x 2x
- 2
x4 + 4x 1
yz
- +
=

f(x
x2
4x
2x
- 4x
+
+
=

=
= C +2

= 2 + 1
23 =+ ]
ii) (x-
z =
c =

o (( + C(2))
z) (E
-
+
2)
q))
2x =

5
2x3 = 2
- +

( +
5&
x3 1 54 6
7
=
.

= :

x = 1

+1 nature
x a
y
:

23
= 12 + 2
I

-
=
2 +

= 3

so , (1 , 3) = 6 (positive so

minimum)
. of
5
(finding equation a function given its derivative)

=
x 5x + C (3 8)
y= - ,

8 =

(43)3 -
5(3) + c

8 = 18 -

15 + C

. 3
8 + C

c = 5

3 5
y=
so, _
5x +
 ="42z
ii) An
i) y
=
y =(
-
=
22
*
= (8)

Gradient of normal : = 85 -
85
= -

2x - = 16 -
8
z
2(4)
-

S
=
- =

-
:

= 4

Equation :

2 4(x 4)
y
= -
-

2 4x 16
y
-
: -

14
y 4x
= -

When x= 0 when 0
y
=

4(0) 14 0 : 4x-14
y
-

14 x 3 5
y
=
-
= .

so Q= (0 -14) ,
Sop =
(3 . 5 , 0)

length = 10 -1-14)2 + 13 5-0)

.

= 14 4 .
i) Gradient of normal at P:

J
· 2
-

:2
4(6 -

2x)

a
-E
=

Equation of normal :

y
-
8 = -

((x -

1)
8 = -
8 +
c

y 8 16
+x t
-
- - + c =

y
Y
= 1

= -

(x 1 +

2y = 17 -
x
y : 2x +
16

When When
a =
0
y =0
2y : k -
0 2(0) = 17 2 -

85 x =
17
y =
.

so 10 8 , .
5) so (17 0)
,

midpoint :

( 854) ,

= (8 . 5 ,
4 .

25)
 ! b

z3 3x2 9x 1 iii) 32.6a

i) y
y
=
-
-
+ =

3x
= 3x2 6n - -
90
0 = 3x2 6x -
-
9 x= - 1 ,
x = 3
x2 - 2x -
3 = 0 -
1243
x2 -
32 + 2 -
3 = 0 "

x(x 3)
(x + 1)(x 3)
-
+

-
1(x 3)
=
-

0
=
c iv)
O
((a3 -
3x2 9x
-
+ 2))dx

x =

point
-1
,

>
-
%

(3
=

,
3

0)
-
:
3
9273
-

0= 33 3(3)2 9(3) + 1
-
-

0 27 k
- -
:
-

27 -
27 + =

1 =
27

33 75
=

-
.

ii) 23 3x2 9x + 2)
y
= -
-

=
( 1)3 3) 1)2 97 1) +27
-
-
-
- -

=
32
max >
-
(- 1 32) ,
i) y =
x(x 1)(x - -

2) Point of intersection C :

y (x2 x)(x
= -
-

2) 1 x - =
2x -
4

Y = x3 -
2x2 - x2 + 2x 1 + 4 =
2x + x

y = x3 -
3x2 + 22 3x = 5

A :
3 - 2 x=
5
Gradient at A (1 :
, 0)
3(1) 6(1) ii) Ax
"Glus-322- 221)de
2
= - + 2 = ,

=
3 -
6 +2
O

-
:
=

Gradient at B (2 0 :
,

6(2) + 2
=
3(2)2 =

=
=
12 -
12 + 2

= 2 = 4 -
8 + 4 - 0

Equation of tangent at A : = O
y 0 -
= -
1(x -

1)
y = -

x + 1 The areas above the x-axis and below the

y :
1 -

x x-axis are
cancelling each other out that
, means

Equation of
tangent at B
: R, and Re are equal
y - 0 2(x 2) = -

y :
2x -
4
Va =

uffy-de y
: 3x

9
y2 =

-9
=
=(6x]
= (6(x) 6(1)]
-

= (48 6]
-

= 425
:
((-2)da 4x
- +
3 = -

1 + 10

4x
y
1 +
++
-

9 4(2)
2 x2 8x x 1 = 0
-

+
-
= + -

x2 + 9x -
14 = 0

9 = 8 -
2 + x2 - 9x + 14 = 0

c = 3 x2-(x -
2x + 14 = 0

3 x(x 7) 2(x 7)
4x
-
0
y
=
-
-
-
:
+

(x 2)(x 3)
- -
= 0

x = =7
2
ii) Gradient of :
normal at P
4
= x
y =
z 10
-

+
-

= 4-2
2 18
-
=
=
+
= -
Equation : =
6 5 .

y -9 =
-

z(x 2) -

Q - ( ,
6 .
5)
y -
9 =
-
Ex + 1

y -
=

Ex +
15
 Sa
+
y
=

2
3x + 1
y
=
-

3x =
y2 -
1

2
x =

= (y2-1)

i)
Ay =Jody ii) Va
= de

= -1 da line y 2 ye 4
: =
,
= curve :
y' Sa +

j4dx =((x
= + 1)dx

-3) S

= (4x] =
5 (1-4) ( 1)
· -
-

(4
(3 G
=
-
0) =

: (5-15)) 45
=

S
:

=
E-Ert == .
4
iii) Y=x+

y = (3x + 1)
-
1) (3)
+(3x
= +

=
-(3x + 1)
2

Gradient at P :
(0 , 1) Gradient at Q :
(1 2)
,

3(3(0)
+
1)-
= (3(1) + 1)

&(1) =
=

tan 0 =
Am

1 + m, mz

z - -
=
()(a)
:
=
tanc = 6
17

0 : 19 40.
 (3x5 6 da 3 6
is ii) 0
-
- =

35 = 6

= 6 E =

4
2

2 x =

=
Y = 2x - 6x + c
3
2 = 2(9)2 -
6(9) +
C =
2xt
2
2 = 54 -

54 + C

=
C = 2

2x2 6x + 2
y
-

i
= 0 .
75 -
positive su minimum point
 Va =
Syndx y=

Chr
= Gain dis
y2
24r = ? (a)
na
y = ax
-
z

240x =

ya)) ( ( +))
- -

24 = 2a ?
3

a? = 36
a = 6
 find limits
y (x -2)"
2) D 2x =
: :
, y+
7 2x
y
= -

(x 2) -
= 7 -
2x

x2 -
4x + 4 = 7 32-

x2 -

2x -
3 =
0

x -
3x 2
+ -

3 = 0

(x + 1)(x 3) -
= 0

x =
-
1 ,
x =
3

Total area : Area under the curve:

fla-2)
·
da
((7-2x)d
- (
=(x x2)
-
-

=
((3) -
9) -

(3) 1)
- -
( - 1)2)

( 8) 1 - 9)
(
= 21 -
9 -
- =
-

=
20

28
=

20-28

= = 10
 j 4

i)x +
4 = 1-4x2ii) y 5, :25
=
= 5
="
1 1 va
G25dn
= - x

= 5 -
x 22 I
2
1 -
1 =
0 = [25x] xo
22
4 = 5x -
22 =
25(4) -

25(1) x -+

75s
x2 5x 4 0 1
5
+ = =
- =

22 -x -
4x + 4 = 0 y : n + y2 ,
= x2 + 8 + 16x22

x(x -

1) 4(x 1)
-
-
= 0
22 = 4

(x 4)(x 1) 0 2 Vx
((a2 8 16x 2)dx
-
+
-
- =
x = = +

x 4 x = 1
y x
= = +
,

=8
x
A : (1 , 5) B :
(4 , 5)
y 1y
x
1 4
= + = x + = 2 +

=
N

1 4 4
4 1
= + = + =

(( 4)]
5

(4) (
M + (2 , 4) = - 8(4) + 8(1)
+
- - -

= 5 = 5

=
( 2) +

= 575

shaded :

75r-57
-
56 5 .
or 18
i) 3x2 + 2n - 5 > 0

3x2 + 2x -
5 =
0

5
-

3x2 3x -
+ 5x -
5 =
0

3x(x -

1) + 5(x -
1) = 0

(3x + 5) =
0x -
1 =
0

x
-5 x 1
= =

x
5 or ses
-

ii)
(132 + 2x-5)dx

=
y
= 23 + x2 -
5x + c (1, 3)

3 = 13 + 12 - 5(1) + 2

3 =
1 + 1 -
5 + C

3 = -
3 + C

c = 6

y = 23 + 22 5x + - 6
i) 9 23
.

= iii) Y 9-23 =
iii)
y :
-3
An
=(19.23) da
:2
9x3 26 - = 8

x0 - 923 + 8 = 0
:i x-]
0= 2302 = x + 3x2
+
=

9u
(9(2) 2 (4() 7)
v2 - + 8 =
G =
-
-

3x2
=
u2 u -
8u + 8 = 0

8(u 1)
v(u 1)
14-3
-
- - = 0 =

(u 8)(u 1) 0
- -
=
3x0 = 24

26
=
u = 8 u = 1 = 5 25. = 8

8 = x3 1 = 23 x = 1 41
.

3
y 8x 4)
-

x = 2 x = =
C = 1 .

a= 1 ,
b= 2 Ax
=J8x-3da
]
= i( 32-2]-

( 4(2) 2) ( 4(1) 2)
-
-

-
- -
=

= -
1 +4 = 3
5. 25 -

3 = 2 25.
 Limit

I 5
n= - 1

Y
=-+
x2

Yet
5
x+ 1 =

Vy= Sign - 1) dy
·
+

=
x -
1

= Y +

ii)
Je-I dy Ayy( -

- -

3) = -

+3 5 + +
3)
= -
y
= E 2) Ex - 1) + + - + +

(2 -

2) -

(2 1)
-

- y +
( 55]
=
-
= -

-
=
4 + 5
1
=
=

= 5 24 .
i) y) 2x-1
y
=

=
gy
ii) An
=5J(2-1 de

=
=
3 En= = 2n -
1 =
((2(5)-1 _
9(2x -

1 = 422 -
42 + /
=

18x -

9 = 422 - 42 + 1

422 -
4x1 -
18x + 1 + 9 =
0 = 9 -
0

4x2 -
22x + 10 =
0 = 9

2x2 -
11x +
5 =
0

2x2 -
10x

2x(x 5) 1(x 5)
-
-

-
x + 5
-
= 0

= 8
An
In-de
=

=
(2x 1)(x 5) 0
- - =

x = 2 or =5a = 5
2

= (52
-5) -
(2)
(2)]
·
9
-
= 27
4

= = 2 25
.
= SRx-Ide S
212 2
-
- 6

x2 x C
=
+
y
= =

-63
x2 x + c = 0 - 10 =

32 -
3 +
c = 0

6 9
0 10
q 18 C
+ c =
-

=
-

-
+

c = - 6

C = 2
2
x2 -
x -
6 = 0

x 2 - 3x 6
y : Gx
2x
+
+ - = 0

x(x 3) - +
2(x 3) - =
0

(x + 2)(x 3) 0
-(2)
-
=
=

x = 2 or 2 = 3
-

- 2
= - + 12 + 2
2

105
· :

1-2 , 10)