2025-KC

M2 Please write your information here.

IE(III)

KCITE CLASSROOM
INTEGRATION EXERCISES 2025

MATHEMATICS Extended Module 2

Integration Exercise III
Question-Answer Book

This paper must be answered in English

INSTRUCTIONS

(1) After the announcement of the start of the examination, you should first write
your name, class and class number in the space provided on Page 1.

(2) Attempt ALL questions in this paper. Write your answers in the spaces
provided in the Question-Answer Book. Do not write in the margins. Answers
written in the margin will not be marked.

(3) Graph paper and supplementary answer sheets will be supplied on request.
Write your name, class and class number, mark the question number box
and fasten them with string INSIDE this book.

(4) Unless otherwise specified, all working must be clearly shown.

(5) Unless otherwise specified, numerical answers should be either exact or

correct to 3 significant figures.

(6) No extra time will be given to candidates for writing their information or filling
in the question number boxes after the ‘Time is up’ announcement.

© Kcite Classroom
No Rights Reserved 2025

1
2025-KC-M2 1–1
 1. Evaluate the following integrals.
1
(a)
 x+x 1000000
dx

e2 cos ( ln x ) − sin ( ln x )
(b)
1 x 2
dx

 ln ( x + 1) dx
1
2
(c)
0

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

________________________________________________________________________________________________________
Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

2
2025-KC-M2 1–2
 ________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________
Answers written in the margins will not be marked.

3
2025-KC-M2 1–3
 2. Given that a is a real constant and f : → .
(a) Show that
2a a
(i) If f ( x ) = f ( 2a − x ) ,
0
f ( x ) dx = 2

0
f ( x ) dx
2a
(ii) If f ( x ) = − f ( 2a − x ) ,
0
f ( x ) dx = 0

x
4
sin   ( x − 2)
 2 
(b) Hence or otherwise, evaluate
 0 4x − x2
dx

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

________________________________________________________________________________________________________
Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

4
2025-KC-M2 1–4
 3. (a) Show that
 
(i)
0
2 ln
(sin x ) dx =  0
2 ln
( cos x ) dx


(ii)
0
ln ( sin x ) dx = 2
0
2 ln
(sin x ) dx

(b) Hence or otherwise, evaluate


(i)
0
2 ln
( sin x ) dx

(ii)
0
4 ln
( tan 2x ) dx

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

5
2025-KC-M2 1–5
 ________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

6
2025-KC-M2 1–6
 ________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________
Answers written in the margins will not be marked.

7
2025-KC-M2 1–7
 4. Given that g and h are real functions defined on the interval − a, a  , where a is some real constant.
(a) Given that g ( x ) is an even function and h ( x ) is an odd function.
a g ( x) a
(i) Show that
 − a e h( x ) +1
dx =
 0
g ( x ) dx

g ( x)e ( )
a k x a
(ii) Let k ( x ) be an odd function also defined on − a, a  , show that

− a eh( x ) +e
k ( x)
dx =

0
g ( x ) dx .

(b) Let f ( x ) = x3 − 9 x 2 + 29 x − 33 .
(i) Show that x − 3 is a factor of f ( x ) .
(ii) If f ( x ) = ( x − 3) Q ( x ) , find the x-coordinate of Q ( x ) .
(iii) Hence, express f ( u + 3) in terms of u .
3
−9 x 2 + 29 x −33
6 ex
(c) Evaluate
 (x
0 2
)(
− 6 x + 18 esin  x + e x
3
−9 x 2 + 29 x −33
)
dx

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

8
2025-KC-M2 1–8
 ________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________
Answers written in the margins will not be marked.

9
2025-KC-M2 1–9
 ________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________
Answers written in the margins will not be marked.

10
2025-KC-M2 1–10
 
5. Let I n =

0
2 sin n x dx , where n  +
.

(a) (i) Find I 0 and I1 .

n −1
(ii) Using integration by parts, show that I n = I n−2 .
n
(b) Hence or otherwise, show that

( 2 m )! 
(i)
0
2 sin 2 m x dx =
( 2 m!) 2
m
2

( 2 m!)
2
 m
(ii)
0
2 sin 2 m +1 x dx =
( 2m + 1)!
n +1
  1
(c) (i) Show that I 0 I1I 2 I 2 n +1 =   for all n  , where for all positive integers m ,
2 ( 2n + 1)!! 0

( 2m )!! = 2  4  6  8   ( 2m − 2 )  ( 2m )
( 2m + 1)!! = 1 3  5  7   ( 2m − 1)  ( 2m + 1)

(iii) Hence or otherwise, evaluate

           
  
 2 sin 7 x dx  2 sin8 x dx  2 sin 9 x dx 
 
 2 sin19 x dx  2 sin 20 x dx 

Answers written in the margins will not be marked.

 0     0  
Answers written in the margins will not be marked.

 0  0  0
      

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

11
2025-KC-M2 1–11
 ________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________
Answers written in the margins will not be marked.

12
2025-KC-M2 1–12
 ________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________
Answers written in the margins will not be marked.

13
2025-KC-M2 1–13
 ________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

________________________________________________________________________________________________________

END OF PAPER
Answers written in the margins will not be marked.

14
2025-KC-M2 1–14