Integration — Paper 2 — A Level Mathematics 9709 topical questions

Question 1

4 18 −5 Show that dx = ln 25. [6] x 2x + 1 1

Accepted Answer

5 Integrate and state term ln x B1 Obtain term of the form kln (2x + 1) M1 State correct term –2ln (2x + 1) A1 Substitute limits correctly M1 Use law for the logarithm of a product, quotient or power M1 Obtain given answer correctly A1 [6] − 1 x 1 x

Question 2

3 Show that 2e dx 12e2 2. [5] ã = + −3 0 (ex + 1)2

Accepted Answer

1 2 x 3 Integrate and obtain e term B1 2 Obtain 2e x term B1 Obtain x B1 Use limits correctly, allow use of limits x = 1 and x = 0 into an incorrect form M1 Obtain given answer A1 [5] S. R. Feeding limits into original integrand, 0/5 dx 1 dy 2

Question 3

3 Show that 2e dx 12e2 2. [5] ã = + −3 0 (ex + 1)2

Accepted Answer

1 2 x 3 Integrate and obtain e term B1 2 Obtain 2e x term B1 Obtain x B1 Use limits correctly, allow use of limits x = 1 and x = 0 into an incorrect form M1 Obtain given answer A1 [5] S. R. Feeding limits into original integrand, 0/5 dx 1 dy 2

Question 4

2 5 2 Show that dx ln 3. [5] 4x 1 = ä 2 +

Accepted Answer

2 Integrate and obtain term of the form kln(4x + 1) M1 1 State correct term ln( 4 x + )1 A1 2 Substitute limits correctly M1 Use law for the logarithm of a quotient or a power M1 Obtain given answer correctly A1 [5] 1 2

Question 5

(i) By ﬁrst expanding show that cos(2x + x), cos 3x cos3x cos x. ≡4 −3 [5] (ii) Hence show that 16π cos3x dx 12.5 ã 0 (2 −cos x) = [5]

Accepted Answer

8 (i) Make relevant use of the cos(A + B) formula M1* Make relevant use of the cos 2A and sin 2A formulae M1* Obtain a correct expression in terms of cos x and sin x A1 Use sin2 x = 1 – cos2 x to obtain an expression in terms of cos x M1(dep*) Obtain given answer correctly A1 [5] 1 1 (ii) Replace integrand by cos 3 x + cos x , or equivalent B1 2 2 1 1 Integrate, obtaining sin 3 x + sin x , or equivalent B1 + B1√ 6 2 Use limits correctly M1 Obtain given answer A1 [5]

Question 6

(i) Show that sin x cos can be written in the form 5 2 sin 2x cos 2x. [5] 2 2 (2 + x)2 + −3 14π (ii) Hence ﬁnd the exact value of sin x cos…

Accepted Answer

7 (i) Expand to obtain 4 sin2 x + 4 sin x cos x + cos2 x B1 Use 2 sin x cos x = sin 2x B1 Attempt to express sin2 x or cos2 x (or both) in terms of cos 2x M1 Obtain correct 12 k 1( − cos 2 x ) for their k sin2 x or equivalent A1√ Confirm given answer 52 + 2 sin 2 x − 32 cos 2 x A1 [5] (ii) Integrate to obtain form px + q cos 2x + r sin 2x M1 Obtain 52 x − cos 2 x − 34 sin 2 x A1 Substitute limits in integral of form px + q cos 2x + r sin 2x and attempt simplification DM1 Obtain 85 π + 14 or exact equivalent A1 [4]

Question 7

x6 (a) Find 4e−1 dx. [2] ã 3 6 (b) Show that dx ln 16. [5] 3x = ä1 −1

Accepted Answer

6 (a) Obtain integral ke 2 with any non-zero k M1 Correct integral A1 [2] (b) State indefinite integral of the form k ln (3x – 1), where k = 2, 6 or 3 M1 State correct integral 2 ln (3x – 1) A1 Substitute limits correctly (must be a function involving a logarithm) M1 Use law for the logarithm of a power or a quotient M1 Obtain given answer correctly A1 [5] dy 2

Question 8

(a) Find 4 cos2 1 [3] Ó 21 d1. 6 1 (b) Find the exact value of dx. [4] Ô 2x 3 −1 +

Accepted Answer

3 (a) Express integrand in the form p cos θ + 2 M1 State correct 2 cos θ + 2 A1 Integrate to obtain 2 sin θ + 2θ (+ c) A1 [3] (b) Integrate to obtain form k ln (2 x + 3) M1 1 Obtain correct ln (2 x + 3) A1 2 Apply limits correctly DM1 1 Obtain ln 15 A1 [4] 2

Question 9

(a) Find tan2x sin 2x dx. [3] Ó + 1 (b) Find the exact value of dx. [4] Ó 0 3e1−2x

Accepted Answer

5 (a) Use tan 2 x = sec 2 x − 1 B1 Obtain integral of form p tan x + qx + r cos 2 x M1 1 Obtain tan x − x − cos 2 x + c A1 [3] 2 (b) Obtain integral of form k1e− 2 x M1* 3 − 2 x Obtain − 1e A1 2 Apply both limits the correct way round M1 dep 3 −1 3 Obtain − e + e or exact equivalent A1 [4] 2 2

Question 10

a 3 Given that 4e Ó0 12x+3 dx = 835, ﬁnd the value of the constant a correct to 3 signiﬁcant ﬁgures. [5] ..................................…

Accepted Answer

3 Integrate to obtain form 1 2 3 + x ke where k is constant not equal to 4 M1 Obtain correct 1 2 3 8 + x e A1 Allow unsimplified for A1 Obtain 1 2 3 3 8 8 835 + − = a e e or equivalent A1 Carry out correct process to find a from equation of form 1 2 3 + = a ke c M1 Obtain 3.65 A1 If 3.65 seen with no actual attempt at integration, award B1 if it is thought that trial and improvement with calculator has been used. Total: 5

Question 11

(a) Find 2 cos cos 1 [4] Ó 1 −3 1 + d1. ...................................................................................................…

Accepted Answer

7(a) 2 2cos cos 3 d θ θ θ − − ∫ B1 Attempt use of identity to obtain integrand involving cos2θ and cosθ M1 Integrate to obtain form 1 2 3 sin2 sin k k k θ θ θ + + for non-zero constants M1 Obtain 1 2 sin2 sin 2 c θ θ θ − − + A1 Total: 4 Question Answer Marks Guidance 7(b)(i) Integrate to obtain form ( ) ( ) 1 2 ln 2 1 ln k x k x + + or ( ) ( ) 1 2 ln 2 1 ln 2 k x k x + + M1 Obtain ( ) 1 2 2ln 2 1 ln x x + + or ( ) ( ) 1 2 2ln 2 1 ln 2 x x + + A1 Total: 2 7(b)(ii) Use relevant logarithm power law for expression obtained from application of limits M1 Use relevant logarithm addition / subtraction laws M1 Obtain ln18 A1 Total: 3

Question 12

(a) Find 2 cos cos 1 [4] Ó 1 −3 1 + d1. ...................................................................................................…

Accepted Answer

7(a) 2 2cos cos 3 d θ θ θ − − ∫ B1 Attempt use of identity to obtain integrand involving cos2θ and cosθ M1 Integrate to obtain form 1 2 3 sin2 sin k k k θ θ θ + + for non-zero constants M1 Obtain 1 2 sin2 sin 2 c θ θ θ − − + A1 Total: 4 Question Answer Marks Guidance 7(b)(i) Integrate to obtain form ( ) ( ) 1 2 ln 2 1 ln k x k x + + or ( ) ( ) 1 2 ln 2 1 ln 2 k x k x + + M1 Obtain ( ) 1 2 2ln 2 1 ln x x + + or ( ) ( ) 1 2 2ln 2 1 ln 2 x x + + A1 Total: 2 7(b)(ii) Use relevant logarithm power law for expression obtained from application of limits M1 Use relevant logarithm addition / subtraction laws M1 Obtain ln18 A1 Total: 3

Question 13

4 (a) Find [4] + sin21 Ô 1 d1. −sin21 .....................................................................................................…

Accepted Answer

4(a) Obtain integrand of form a sec 2 θ+ b M1 Obtain correct 5sec 2 θ− 1 A1 Integrate to obtain form a tanθ+ bθ M1 Obtain 5tanθ− θ+ c A1 4 4(b) Obtain integral of form k ln(3 x + 1) *M1 Apply limits and obtain 23 ln(3a + 1) = ln16 A1 Obtain equation with no presence of ln DM1 Obtain 21 A1 4

Question 14

406 (a) Find the exact value of sin x 4 sin x 6 cosx dx. [5] Ó 0 + ........................................................................…

Accepted Answer

6(a) Obtain 2 − 2cos2x as part of integrand B1 Obtain 3sin 2x as part of integrand B1 Allow second B1 for writing Integrate to obtain form M1  1 2  , M1 6sin x cos x d x = 6  sin x  k1 x + k 2 sin 2 x + k 3 cos2 x ∫  2  may then be implied by subsequent work Obtain 2 x − sin2 x − 32 cos2 x or A1 2 x − sin 2 x + 3sin 2 x Apply limits to obtain 12 π+ 12 A1 5 6(b) B1 6 Integrate to obtain 2ln(3 x + 2) Allow ln ( 3 x + 2 ) for B1 3 Use at least one relevant logarithm property *M1 3a + 2 (3a + 2) 2 A1 Obtain = 7 or = 49 or 2 4 equivalent without ln Solve relevant equation to find a DM1 Dep on *M1, allow for 49 = ( 3a + 2 ) 2 OE or correct working involving ( 3a + 2 ) Obtain a = 4 only A1 5

Question 15

4 (a) Find [4] + sin21 Ô 1 d1. −sin21 .....................................................................................................…

Accepted Answer

4(a) Obtain integrand of form a sec 2 θ+ b M1 Obtain correct 5sec 2 θ− 1 A1 Integrate to obtain form a tanθ+ bθ M1 Obtain 5tanθ− θ+ c A1 4 4(b) Obtain integral of form k ln(3 x + 1) *M1 Apply limits and obtain 23 ln(3a + 1) = ln16 A1 Obtain equation with no presence of ln DM1 Obtain 21 A1 4

Question 16

a 1 2x 26 It is given that 1 e dx 10, where a is a positive constant. Ó 0 + = P Q 15 (i) Show that a 2 ln −a1 . [6] = 2a 4 e + ............…

Accepted Answer

6(i) Rewrite integrand as 1 2 1 2e e + + x B1 Integrate to obtain form 1 2 1 2 e e + + x x x k k M1 Obtain 1 2 4e e + + x x x A1 Use limits to obtain 1 2 4e e 5 10 + + − = a a a A1 Rearrange as far as 1 2e ... = a including use of 1 1 1 2 2 2 4e e e (4 e ) + = + a a a a M1 Confirm 1 2 15 2ln 4 e   − =   +   a a a A1 AG; necessary detail needed 6 6(ii) Consider sign of 1 2 15 2ln 4 e   − −   +   a a a for 1.5 and 1.6 or equivalent M1 Obtain 0.08... − and 0.06… or equivalents and justify conclusion A1 2 6(iii) Use iterative process correctly at least once M1 Obtain final answer 1.56 A1 Show sufficient iterations to 5 sf to justify answer or show sign change in interval (1.555,1.565) A1 3

Question 17

6 2 Show dx ln 125. [5] 2x 1 = thatÔ1 + ...................................................................................................…

Accepted Answer

2 Integrate to obtain form ln(2 1) k M1 Obtain correct 3ln(2 1) x + A1 Use subtraction law of logarithms correctly M1 Dependent on first M1 Use power law of logarithms correctly M1 Dependent on first M1 Confirm ln125 A1 5

Question 18

(a) Find tan2 3x dx. [3] Ó ................................................................................................................…

Accepted Answer

4(a) Use identity 2 2 tan 3 sec 3 1 x x = − B1 Integrate to obtain form 1 2 tan3 k x k x + M1 Obtain correct 1 3 tan3x x c − + A1 3 4(b) Express integrand as 2 e 4e x x − + B1 Integrate to obtain form 2 3 4 e e x x k k − + M1 Obtain correct 2 1 2 e 4e x x − − A1 Use limits to obtain 2 1 7 1 2 2 e 4e− − + or similarly simplified equivalent A1 4

Question 19

(a) Show that 3 sin cot 6 [2] 21 1  cos21. ...............................................................................................…

Accepted Answer

8(a) Use at least one of sin 2 2sin cos θ θ θ = and cos cot sin θ θ θ = B1 Use both and conclude 2 6cos θ AG B1 2 8(b) Attempt solution of 2 5 cos 6 θ = to find at least one value M1 Obtain 0.421 A1 Obtain 2.72 A1 3 Question Answer Marks 8(c) Express integrand in form cos + a b x M1 Obtain correct integrand 3 3cos + x A1 Integrate to obtain sin + px q x *M1 Apply limits correctly DM1 Obtain 3 3 3 4 2 π + − or exact equivalent A1 5

Question 20

(a) Show that 3 sin cot 6 [2] 21 1  cos21. ...............................................................................................…

Accepted Answer

8(a) Use at least one of sin 2 2sin cos θ θ θ = and cos cot sin θ θ θ = B1 Use both and conclude 2 6cos θ AG B1 2 8(b) Attempt solution of 2 5 cos 6 θ = to find at least one value M1 Obtain 0.421 A1 Obtain 2.72 A1 3 Question Answer Marks 8(c) Express integrand in form cos + a b x M1 Obtain correct integrand 3 3cos + x A1 Integrate to obtain sin + px q x *M1 Apply limits correctly DM1 Obtain 3 3 3 4 2 π + − or exact equivalent A1 5

Question 21

(a) Express 5 3 cos x 5 sin x in the form R cos x , where R 0 and 0 1 [3] + −! > < ! < 2π. ................................................…

Accepted Answer

7(a) State 10 R = B1 Use appropriate trigonometry to find α M1 Obtain 1 6 α = π A1 3 7(b) State 6 − B1 FT Following their value of R. Attempt to find x from their cos( ) 1 x α − = − M1 Obtain 1 6 x − π = −π and hence 5 π 6 − A1 3 7(c) State integrand of form 2 1 1 sec 3 π 6 k θ   −     *M1 Integrate to obtain form 2 1 tan 3 π 6 k θ   −     DM1 Obtain 1 1 tan 3 π 300 6 c θ   − +     A1 3

Question 22

1 Find the exact value of 4e2x dx. [4] Ó −2e−x −1 .........................................................................................…

Accepted Answer

1 Integrate to obtain 2 Integrate to obtain 2e−x B1 Apply limits correctly to integral of the form 2 1 2 e e− + x x k k M1 1 4 ≠ k . Condone one error. Obtain 4 2e 2e − A1 or exact equivalent. 4

Question 23

(a) By ﬁrst expanding cos 2 , show that cos 3 4 cos3 3 cos . [3] ..........................................................................…

Accepted Answer

7(a) θ θ θ θ − B1 Attempt correct relevant identities to express in terms of cosθ only M1 M0 if moving terms from side to side. Confirm 3 4cos 3cos θ θ − with sufficient detail A1 AG 3 7(b) Use identity with 5 18 π θ = M1 Obtain 5 1 2 6 cos π and hence 1 4 3 − A1 2 7(c) Express integrand in form 1 2 (cos3 3cos ) (cos9 3cos3 ) + + + k x x k x x M1 Obtain correct integrand 9cos cos9 − x x A1 OE (allow unsimplified). Integrate to obtain form 3 4 sin sin9 + k x k x M1 Obtain correct 1 9 9sin sin9 − x x A1 Now simplified; condone missing ... + c . 4

Question 24

(a) Show that 4 sin 1 cos 3 2 sin [4] 1 + 3π 1 −13π  + 21. ...............................................................................…

Accepted Answer

6(a) Obtain at least either 1 1 2 2 ( sin 3cos )    or 1 1 2 2 ( cos 3sin )    Expand and simplify with correct use of 2 2 sin cos 1     M1 Use 1 2 sin cos sin 2     M1 Confirm given result 3 2sin2  A1 AG necessary detail required. 4 6(b) Identify value of  is 3 8 π *B1 OE Obtain 3 4 3 2sin π  and conclude 3 2  DB1 or exact equivalent. 2 6(c) Identify integrand as 3 2sin4  x B1 Integrate to obtain form 1 2 cos4  k x k x M1 where 1 2 0  k k . Obtain correct 1 2 3 cos4  x x A1 Obtain 1 1 8 2 π 3  A1 or exact equivalent. 4

Question 25

(a) Show that 4 sin 1 cos 3 2 sin [4] 1 + 3π 1 −13π  + 21. ...............................................................................…

Accepted Answer

6(a) Obtain at least either 1 1 2 2 ( sin 3cos )    or 1 1 2 2 ( cos 3sin )    B1 Allow if implied by decimal values. Expand and simplify with correct use of 2 2 sin cos 1     M1 Use 1 2 sin cos sin 2     M1 Confirm given result 3 2 sin 2  A1 AG necessary detail required. 4 6(b) Identify value of  is 3 8 π *B1 OE Obtain 3 4 3 2sin π  and conclude 3 2  DB1 or exact equivalent. 2 6(c) Identify integrand as 3 2 sin 4  x B1 Integrate to obtain form 1 2 cos 4  k x k x M1 where 1 2 0  k k . Obtain correct 1 2 3 cos4  x x A1 Obtain 1 1 8 2 π 3 A1 or exact equivalent. 4

Question 26

(a) Find the quotient when 6x3 is divided by 2x 1 , and show that the remainder is 6. −5x2 −24x −4 + [3] ..................................…

Accepted Answer

5(a) Carry out division at least as far as 3x 2 + k1 x M1 OE (e.g. by inspection). Obtain quotient 3 x 2 − 4 x − 10 A1 Confirm given result of remainder is 6 with sufficient detail A1 AG SC If remainder = 6 shown using remainder theorem allow B1. 3 5(b) Integrate to obtain at least 3x and term of form k 2 ln(2 x + 1) *M1 ln term must be added. Obtain x 3 − 2 x 2 − 10 x + 3ln(2 x + 1) A1 Apply limits correctly to expression with four terms DM1 Apply appropriate logarithm properties correctly to obtain the form k3 ln a DM1 Obtain 195 + ln27 A1 5

Question 27

(a) Find dx, giving your answer in the form ln a, where a is an integer. [4] 2x 4 −5 ......................................................…

Accepted Answer

3(a) Obtain 2ln(2 x − 5) B1 Apply limits correctly M1 For integral of form k ln(2 x − 5) . Use one relevant logarithm property correctly M1 For integral of form k ln(2 x − 5) . Apply second logarithm property correctly and obtain ln25 A1 4 3(b) 1 2 x− 5 B1 Integrate to obtain e 2 1 15 1 3 B1FT or exact equivalent, FT on their ke 2 x −.5 Obtain final answer e − e 2 2 2

Question 28

(a) Prove that 2 sin i cosec 2 i / sec i. [2] .............................................................................................…

Accepted Answer

7(a) 1 cosec2 sin 2    M1 Obtain 1 cos and confirm sec A1 Answer given – necessary detail needed. 2 7(b) Attempt to obtain quadratic equation in sec or cos only *M1 Obtain 2 7 2 sec 1 sec 8     involving one trigonometric ratio A1 Or equivalent, may be unsimplified, but reduce to 2 2sec 7sec 18 0      2 18cos 7cos 2 0      . Attempt to solve 3-term quadratic equation for sec, using a correct method, to find at least one value of  DM1 Or equivalent using cos. Obtain any two of the four correct solutions 0.952, 1.76   A1 Or greater accuracy. Obtain remaining two correct solutions A1 Or greater accuracy; and no others between π  and π . 5 7(c) Identify integrand as 2 1 2 2sec x B1 Integrate 2 1 2 sec k x to obtain 1 2 2 tan k x M1 Obtain correct 1 2 4tan x A1 Condone omission of ...c . 3

Question 29

(a) Prove that 2 sin i cosec 2 i / sec i. [2] .............................................................................................…

Accepted Answer

7(a) 1 cosec2 sin 2    M1 Obtain 1 cos and confirm sec A1 Answer given – necessary detail needed. 2 7(b) Attempt to obtain quadratic equation in sec or cos only *M1 Obtain 2 7 2 sec 1 sec 8     involving one trigonometric ratio A1 Or equivalent, may be unsimplified, but reduce to 2 2sec 7sec 18 0      2 18cos 7cos 2 0      . Attempt to solve 3-term quadratic equation for sec, using a correct method, to find at least one value of  *DM1 Or equivalent using cos. Obtain any two of the four correct solutions 0.952, 1.76   A1 Or greater accuracy. Obtain remaining two correct solutions A1 Or greater accuracy; and no others between π  and π . 5 7(c) Identify integrand as 2 1 2 2sec x B1 Integrate 2 1 2 sec k x to obtain 1 2 2 tan k x M1 Obtain correct 1 2 4tan x A1 Condone omission of ...c . 3

Question 30

a 10 5 It is given that d x = 7 , where a is a constant greater than 1. y 2x + 1 a (a) Show that a = 3 0.5e 1 .4 ( 2 a + 1 ) - 0. 5 . [5] .…

Accepted Answer

5(a) Obtain integral of form k ln(2 x + 1) *M1 Obtain correct 5ln(2 x + 1) A1 Apply limits correctly and equate to 7 DM1 Apply appropriate logarithm property to reach at least a 3 = ... DM1 3 1.4 A1 AG – necessary detail needed. Confirm a = 0.5e (2a + 1) − 0.5 5 5(b) Use iterative process correctly at least once M1 Obtain final answer 2.18 A1 Answer required to exactly 3 sf. Show sufficient iterations to 5 sf to justify answer or show a sign change in the A1 interval [2.175, 2.185] 3

Question 31

(a) Find the quotient and remainder when 18x 3 - 6 x 2 - 30x + 4 is divided by ( 3x - 1 ) . [3] ...........................................…

Accepted Answer

6(a) Carry out division at least as far as 6 x 2 + k1 M1 Obtain quotient 6 x 2 − 10 A1 Obtain remainder − 6 A1 3 6(b) 2 6 B1 FT Following their quotient and remainder. Identify integrand as 6 x − 10 − 3 x − 1 Integrate to obtain at least 2x 3 and term of the form k 2 ln(3 x − 1) *M1 Obtain 2 x 3 − 10 x − 2ln(3 x − 1) A1 FT Following their quotient and remainder. Apply limits and appropriate logarithm properties DM1 Obtain 208 − ln49 A1 5

Question	Answer	Marks	From
1	see sheet	6	9709/21 Oct/Nov 2008
2	see sheet	5	9709/21 Oct/Nov 2010
3	see sheet	5	9709/22 Oct/Nov 2010
4	see sheet	5	9709/21 Oct/Nov 2011
5	see sheet	10	9709/21 Oct/Nov 2011
6	see sheet	9	9709/21 May/June 2012
7	see sheet	7	9709/22 Oct/Nov 2012
8	see sheet	7	9709/21 Oct/Nov 2014
9	see sheet	7	9709/21 Oct/Nov 2015
10	see sheet	5	9709/21 May/June 2017
11	see sheet	9	9709/22 May/June 2017
12	see sheet	9	9709/23 May/June 2017
13	see sheet	8	9709/21 Oct/Nov 2017
14	see sheet	10	9709/22 Oct/Nov 2017
15	see sheet	8	9709/23 Oct/Nov 2017
16	see sheet	11	9709/22 May/June 2018
17	see sheet	5	9709/21 Oct/Nov 2018
18	see sheet	7	9709/21 May/June 2019
19	see sheet	10	9709/22 May/June 2020
20	see sheet	10	9709/23 May/June 2020
21	see sheet	9	9709/22 Feb/March 2021
22	see sheet	4	9709/21 Oct/Nov 2021
23	see sheet	9	9709/23 Oct/Nov 2021
24	see sheet	10	9709/22 May/June 2023
25	see sheet	10	9709/23 May/June 2023
26	see sheet	8	9709/22 Oct/Nov 2023
27	see sheet	6	9709/23 Oct/Nov 2023
28	see sheet	10	9709/22 May/June 2024
29	see sheet	10	9709/23 May/June 2024
30	see sheet	8	9709/23 Oct/Nov 2024
31	see sheet	8	9709/22 Feb/March 2025

Integration — Paper 2 · A Level Mathematics 9709

Questions38 pages

Mark scheme31 answers

Another paper, or another topic

Questions as text

Q1 · 4 18 −5 Show that dx = ln 25 9709/21 Oct/Nov 2008

Q2 · 3 Show that 2e dx 12e2 2 9709/21 Oct/Nov 2010

Q3 · 3 Show that 2e dx 12e2 2 9709/22 Oct/Nov 2010

Q4 · 2 5 2 Show that dx ln 3 9709/21 Oct/Nov 2011

Q5 · By ﬁrst expanding show that cos(2x + x), cos 3x cos3x cos x 9709/21 Oct/Nov 2011

Q6 · Show that sin x cos can be written in the form 5 2 sin 2x cos 2x 9709/21 May/June 2012

Q7 · X6 (a) Find 4e−1 dx 9709/22 Oct/Nov 2012

Q8 · Find 4 cos2 1 [3] Ó 21 d1 9709/21 Oct/Nov 2014

Q9 · Find tan2x sin 2x dx 9709/21 Oct/Nov 2015

Q10 · A 3 Given that 4e Ó0 12x+3 dx = 835, ﬁnd the value of the constant a correct to 3… 9709/21 May/June 2017

Q11 · Find 2 cos cos 1 [4] Ó 1 −3 1 + d1 9709/22 May/June 2017

Q12 · Find 2 cos cos 1 [4] Ó 1 −3 1 + d1 9709/23 May/June 2017

Q13 · 4 (a) Find [4] + sin21 Ô 1 d1 9709/21 Oct/Nov 2017

Q14 · 406 (a) Find the exact value of sin x 4 sin x 6 cosx dx 9709/22 Oct/Nov 2017

Q15 · 4 (a) Find [4] + sin21 Ô 1 d1 9709/23 Oct/Nov 2017

Q16 · A 1 2x 26 It is given that 1 e dx 10, where a is a positive constant 9709/22 May/June 2018

Q17 · 6 2 Show dx ln 125 9709/21 Oct/Nov 2018

Q18 · Find tan2 3x dx 9709/21 May/June 2019

Q19 · Show that 3 sin cot 6 [2] 21 1 cos21 9709/22 May/June 2020

Q20 · Show that 3 sin cot 6 [2] 21 1 cos21 9709/23 May/June 2020

Q21 · Express 5 3 cos x 5 sin x in the form R cos x , where R 0 and 0 1 [3] + −! 9709/22 Feb/March 2021

Q22 · 1 Find the exact value of 4e2x dx 9709/21 Oct/Nov 2021

Q23 · By ﬁrst expanding cos 2 , show that cos 3 4 cos3 3 cos 9709/23 Oct/Nov 2021

Q24 · Show that 4 sin 1 cos 3 2 sin [4] 1 + 3π 1 −13π + 21 9709/22 May/June 2023

Q25 · Show that 4 sin 1 cos 3 2 sin [4] 1 + 3π 1 −13π + 21 9709/23 May/June 2023

Q26 · Find the quotient when 6x3 is divided by 2x 1 , and show that the remainder is 6 9709/22 Oct/Nov 2023

Q27 · Find dx, giving your answer in the form ln a, where a is an integer 9709/23 Oct/Nov 2023

Q28 · Prove that 2 sin i cosec 2 i / sec i 9709/22 May/June 2024

Q29 · Prove that 2 sin i cosec 2 i / sec i 9709/23 May/June 2024

Q30 · A 10 5 It is given that d x = 7 , where a is a constant greater than 1 9709/23 Oct/Nov 2024

Q31 · Find the quotient and remainder when 18x 3 - 6 x 2 - 30x + 4 is divided by ( 3x - 1 ) 9709/22 Feb/March 2025