SM 025 INTEGRATION

1. Find BRAIN BOOSTER 1

(a) ex 1 dx. Ch 1 Integration
ex (2 marks)
(2 marks)
(b) 5 x dx.

2. Evaluate 2 8 sin2 2xdx. (4 marks)
(7 marks)
4 (7 marks)

3. (a) Evaluate 1 x3 dx.
x2 )2
0 (1

(b) 4 x dx.

By using substitution, find the value of
0 5x2 1

4. (a) By using integration by parts, evaluate 6 x cos 6xdx. (6 marks)
(7 marks)
0

(b) Evaluate e ln x2xdx. Give your answer in terms of e.

1

5. Write 18x 9 as a sum of partial fractions.
(x 2)(x 1)2

Hence, find (x 2x 1 1)2 dx. (8 marks)
2)(x

6. (a) The diagram shows the region R bounded by the curve y 3 x ln x and the
lines y 0, x 2, x 8.

x2 x 8 y 3 x ln x

Find the area of R. (7 marks)
(b) Let R be a region bounded by y ex, y 2, y 5 and y-axis.

BRAIN BOOSTER 1

Ch 1 Integration

(i) Find the area of R. (7 marks)

(ii) Hence, find the volume of the solid generated by revolving the region R

through 360 about y-axis. (7 marks)

Answers 3 C

11 2(5 x)2
xx C (b)

1. (a) 2e2 2e 2 3

2. 8
(b)
3. (a) 0.0966
5

1 e2 1
4. (a)
(b)
18
2
5. 5 ln x
9 2 5 ln x 1 1 C

9 3(x 1)

6. (a) 16.0607 units2 (b) (i) 3.6609 units2 (b) (ii) 4.6688 units3

Solution BRAIN BOOSTER 1

NO ANSWER SCHEME Ch 1 Integration
1(a) ex 1
ex 1 MARKS
dx 1x 1x dx
ex e2 e2 K1
J1
1x 1x
K1
e2 e 2 dx
J1
1x 1x 4M
B1
2e2 2e 2 C K1

1(b) 1 K1
(b) 5 x dx J1
(5 x)2 dx 4M
B1
3 J1

(5 x)2 C
3( )
2

3

(5 x)2
C

3

2 2 8 sin2 2xdx 2 8 1 cos 4x dx

4 42

sin 4x 2
4x

4

4

4x sin 4x 2

4

4 sin 4 4 sin 4
2 24 4

3(a)

u 1 x2 x2 u 1

du 2x
dx
du x dx
2

BRAIN BOOSTER 1

Ch 1 Integration

NO ANSWER SCHEME MARKS

1 x3 1 x2(x) dx
dx 0 (1 x2 )2

0 (1 x2 )2

u 1 du K1
u2 2 K1
1 u2
du
2u 2

1 ln u 1 C
2 2u

1 ln 1 x2 1 J1
2
1
2(1 x2)

0

1 ln 1 12 1 1 ln 1 02 1
2 2(1 12) 2 (1 02)
0.0966 K1
J1

u 5x2 1 K1
3(b)

du 10x
dx
du 10x dx

du xdx
10

4 x dx 1 du
0 5x2 1 u 10

11 K1
u 2du K1
K1
10
K1
1 c

1 u2 4
10 1
1
2
0
5x2
5

NO ANSWER SCHEME BRAIN BOOSTER 1

4(a) 5(4)2 1 5(0)2 1 Ch 1 Integration
ux MARKS
55 K1
du 1 J1
dx 8 14 M
du dx 5
J1 J1
dv cos 6xdx
v sin 6x
6

6 x cos 6xdx x sin 6x sin 6x dx K1
06 6 K1
K1
x sin 6x cos 6x 6 J1
6 36 0 J1 J1

sin 6 cos 6 (0)sin 6(0) cos 6(0) K1
66 6 6 36 K1

6 36

1
18

4(b) u ln x dv 2xdx
v 2x2 x2
du 1 2
dx x
du 1 dx

x

e ln x2xdx e

1 2x ln xdx

1

(ln x)x2 x2( 1 dx)
x

x2 ln x xdx

BRAIN BOOSTER 1

Ch 1 Integration

NO ANSWER SCHEME MARKS
K1
x2 ln x x2 e
2 K1

1 J1
13 M
e2 ln e e2 12 ln 1 12 B1
2 2
K1
e2 1

2

5 18x 9 AB C

(x 2)(x 1)2 x 2 x 1 (x 1)2

18x 9 A(x 1)2 B(x 1)(x 2) C(x 2)

When x 1,
18(1) 9 C(1 2)

3C

When x 2, 1)2 K1
18( 2) 9 A( 2
A
5

When x 0, K1
18(0) 9 5(0 1)2 B(0 1)(0 2) 3(0 2)

9 5 2B 6
5B

18x 9 55 3
(x 2)(x 1)2
x 2 x 1 (x 1)2 J1
K1
2x 1 1)2 dx 1 18x 9 dx K1
2)(x 9 2)(x
(x (x 1)2 J1

1 18x 9 dx 1 55 3 dx

9 (x 2)(x 1)2 9 x 2 x 1 (x 1)2

1 5 ln x 2 5 ln x 1 3 C
9 x1

5 ln x 2 5 ln x 1 1 C
9 9 3(x 1)

8M

NO ANSWER SCHEME BRAIN BOOSTER 1
6(a)
1 Ch 1 Integration
u ln x MARKS
dv x3dx J1 J1
du 1
4 3 4 B1
dx x K1
du 1 dx v x3 4 x3
4 K1
x K1
3 J1

8 J1 J1

A 3 x ln xdx B1
K1
2 K1
K1
(ln x) 3 4 8 3 4 1 J1
3
x3 8

4 x ( dx)
24 x
2

3 4 8 3 81

4 x3 ln x 42 x 3 dx

2

3 4 9 4 8

4 x3 ln x 16 x3

2

3 4 ln(8) 9 4 3 4 ln(2) 9 4

(8)3 (8)3 (2)3 (2)3
4 16 4 16

16.0607 units2

(b)(i)

u ln y dv dy
du 1 vy
dy y
du 1 dy

y

5

A ln ydy

2

(ln y)y y( 1 )dy
y

y ln y 1 dy

[y ln y y]52 2)
(5 ln 5 5) (2 ln 2
3.6609 units2

NO ANSWER SCHEME BRAIN BOOSTER 1
(b)(ii)
dv dy Ch 1 Integration
u (ln y)2 vy MARKS
du 2(ln y) 1
dy y J1 J1
du 2 ln y dy
B1
y K1

5 K1

V (ln y)2 dy K1
J1
2 21 M

(ln y)2 y 5 5 2 ln y
2 y dy
2y

(ln y)2 y 5 5
2
2 ln y dy

2

y(ln y)2 5 2 5
22
ln y dy

5(ln 5)2 2(ln 2)2 2 (3.6609)

4.6688 units3