Integration — Paper 3 — A Level Mathematics 9709 topical questions

Question 1

1 Find the exact value of the constant k for which dx = 1. [4] 2x −1 1

Accepted Answer

1 Obtain indefinite integral of the form aln(2x –1), where a = 12 , 1, or 2 M1 Use limits and obtain equation 1 ln ( 2 k − )1 = 1 A1 2 Use correct method for solving an equation of the form aln(2k –1) = 1, where a = 1 , 1, or 2, for k M1 2 Obtain answer k = 1 e( 2 + )1 , or exact equivalent A1 [4] 2 2 2

Question 2

y P x O T N 12 In the diagram the tangent to a curve at a general point P with coordinates (x, y) meets the x-axis at T. The point N on the…

Accepted Answer

y d y 8 (i) State = , or equivalent B1 TN d x dy Express area of PTN in terms of y and , and equate to tan x M1 dx Obtain given relation correctly A1 [3] (ii) Separate variables correctly B1 2 Integrate and obtain term − , or equivalent B1 y Integrate and obtain term ln(sin x), or equivalent B1 Evaluate a constant or use limits y = 2, x = 1 π in a solution containing a term of the 6 form a/y or bln(sin x) M1 2 Obtain correct solution in any form, e.g. − = ln (2 sin x ) − 1 A1 y Rearrange as y = 2 / (1 − ln (2 sin x )) , or equivalent A1 [6] [Allow decimals, e.g. as in a solution y = 2 / (3.0 − ln (sin x )) .]

Question 3

(i) Use the substitution x 2 tan θ to show that = 2 1 8 4π dx cos2 θ dθ. ä 0 [4] 0 = ã (4 + x2)2 (ii) Hence ﬁnd the exact value of 2 8 dx. …

Accepted Answer

dx 6 (i) State or imply = 2sec2 θ or dx = 2 sec2 θ dθ B1 dθ Substitute for x and dx throughout M1 Obtain any correct form in terms of θ A1 Obtain the given form correctly (including the limits) A1 [4] (ii) Use cos 2A formula, replacing integrand by a + b cos 2θ, where ab ≠ 0 M1* Integrate and obtain 1 θ + 1 sin 2θ A1 2 4 Use limits θ = 0 and θ = 1 π M1(dep*) 4 Obtain answer 1 (π + 2), or exact quivalent A1 [4] 8

Question 4

2x 7 5 Show that dx ln 50. [7] + ä 0 = (2x + 1)(x + 2)

Accepted Answer

A B 5 State or imply form + B1 2 x + 1 x + 2 Use relevant method to find A or B M1 4 1 Obtain − A1 2 x + 1 x + 2 Integrate and obtain 2 ln (2 x + 1) − ln ( x + 2 ) (ft on their A, B) B1√B1√ Apply limits to integral containing terms a ln (2 x + )1 and b ln ( x + 2 ) and apply a law of logarithms correctly. M1 Obtain given answer ln 50 correctly A1 [7]

Question 5

7 The integral I is deﬁned by I 4t3 dt. = ã 0 ln(t2 + 1) 5 (i) Use the substitution x t2 1 to show that I ln x dx. [3] = + = ã 1 (2x −2) (i…

Accepted Answer

7 (i) State or imply dx = 2t dt or equivalent B1 Express the integral in terms of x and dx M1 5 Obtain given answer ∫ ( 2 x − 2) ln x d x , including change of limits AG A1 [3] 1 2 1 (ii) Attempt integration by parts obtaining (ax2 + bx)ln x ±∫ ( ax + bx ) x dx or equivalent M1 2 1 Obtain (x2 – 2x)ln x –∫ ( x − 2 x ) x dx or equivalent A1 Obtain (x2 – 2x)ln x – 12 x2 + 2x A1 Use limits correctly having integrated twice M1 Obtain 15 ln 5 – 4 or exact equivalent A1 [5] [Equivalent for M1 is (2x – 2)(ax ln x + bx) –∫ ( ax ln x + bx ) 2dx]

Question 6

2x 23 Show that 4e −1 [5] dx = −2. ã 0 (1 −x)e−1

Accepted Answer

x x , or equivalent M13 Attempt integration by parts and reach k (1 − x )e 2 ± k ∫ e 2 d 1 1 − x − x x , or equivalent A1 Obtain − 2(1 − x )e 2 − 2 ∫ e 2 d 1 1 − x − x Integrate and obtain − 2(1 − x )e 2 + 4 e 2 , or equivalent A1 Use limits x = 0 and x = 1, having integrated twice M1 Obtain the given answer correctly A1 [5]

Question 7

y M 1 x O 2p The diagram shows the curve y 5 sin3x cos2x for 0 2π, and its maximum point M. = ≤x ≤1 (i) Find the x-coordinate of M. [5] (ii…

Accepted Answer

8 (i) Use product and chain rule M1 Obtain correct derivative in any form, e.g. 15 sin 2 x cos 3 x − 10 sin 4 x cos x A1 Equate derivative to zero and obtain a relevant equation in one trigonometric function M1 Obtain 2 tan 2 x = 3 , 5 cos 2 x = 2 , or 5 sin 2 x = 3 A1 Obtain answer x = 0.886 radians A1 [5] du (ii) State or imply d u = − sin x d x , or = − sin x , or equivalent B1 dx Express integral in terms of u and du M1 Obtain ± 5(u 2 − u 4 ) ∫ d u , or equivalent A1 1 Integrate and use limits u = 1 and u = 0 (or x = 0 and x = π ) M1 2 2 Obtain answer , or equivalent, with no errors seen A1 [5] 3 dx ( )( )

Question 8

(i) Use the substitution u tan x to show that, for n = ≠−1, 14π 1 dx ã 0 (tann+2x + tannx) = n 1. [4] + (ii) Hence ﬁnd the exact value of 1…

Accepted Answer

du 10 (i) State or imply = sec 2 x B1 dx Express integrand in terms of u and du M1 u n +1 Integrate to obtain or equivalent A1 n + 1 1 Substitute correct limits correctly to confirm given result A1 [4] n + 1 (ii) (a) Use sec2 x =1 + tan2 x twice M1 Obtain integrand tan4 x + tan2 x A1 Apply result from part (i) to obtain 13 A1 [3] Or M1 Use sec2 x = 1 + tan2 x and the substitution from (i) Obtain ∫ u2du A1 Apply limits correctly and obtain 13 A1 (b) Arrange, perhaps implied, integrand to B1 t9 + t7 + 4(t7 + t5) + t5 + t3 Attempt application of result from part (i) at least twice M1 1 4 1 25 Obtain + + and hence 24 or exact equivalent A1 [3] 8 6 4

Question 9

8 (a) Show that 4x ln x dx 56 ln 2 [5] Ó2 = −12. 1 240 (b) Use the substitution u sin 4x to ﬁnd the exact value of cos34x dx. [5] = Ó 0

Accepted Answer

8 (a) Carry out integration by parts and reach ax 2 ln x + b ∫ 12 x 2 d x M1* Obtain 2 x 2 ln x −∫ 1x . 2 x 2 d x A1 Obtain 2 x 2 ln x − x 2 A1 Use limits, having integrated twice M1 (dep*) Confirm given result 56ln 2 − 12 A1 [5] GCE AS/A LEVEL – May/June 2013 9709 31 (b) State or imply ddux = 4cos4 x B1 Carry out complete substitution except limits M1 Obtain ∫ ( 14 − 14 u 2 ) d u or equivalent A1 Integrate to obtain form k1u + k 2 u 3 with non-zero constants k1 , k 2 M1 Use appropriate limits to obtain 1196 A1 [5]

Question 10

(i) Prove that cot tan 2 cosec [3] 1 + 1  21. 1 30 1 (ii) Hence show that cosec ln 3. [4] 1 2 Ó 21 d1 = 60

Accepted Answer

5 (i) Use Pythagoras M1 Use the sin2A formula M1 Obtain the given result A1 [3] (ii) Integrate and obtain a k ln sin θ or m ln cos θ term, or obtain integral of the form p ln tan θ M1* 1 1 1 Obtain indefinite integral ln sin θ − ln cos θ , or equivalent, or ln tan θ A1 2 2 2 Substitute limits correctly M1(dep)* Obtain the given answer correctly having shown appropriate working A1 [4] 2 2 2 ( )

Question 11

(i) Prove that cot + tan 2 cosec 2 . [3] 1 3 1 (ii) Hence show that cosec 2 d = ln 3. [4] 1 2 6

Accepted Answer

5 (i) Use Pythagoras M1 Use the sin2A formula M1 Obtain the given result A1 [3] (ii) Integrate and obtain a k ln sin θ or m ln cosθ term, or obtain integral of the form p ln tan θ M1* 1 1 1 Obtain indefinite integral ln sin θ − ln cos θ , or equivalent, or ln tan θ A1 2 2 2 Substitute limits correctly M1(dep)* Obtain the given answer correctly having shown appropriate working A1 [4] 2 2 2 ( )

Question 12

2 Find the exact value of dx. [5] Ó 0 xe−2x

Accepted Answer

2 Integrate by parts and reach axe −2 x + b ∫ e −2 x dx M1 Obtain − 12 xe −2 x + 12 ∫ e −2 x dx , or equivalent A1 Complete the integration correctly, obtaining − 12 x e − 2 x − 14 e − 2 x , or equivalent A1 Use limits x = 0 and x = 12 correctly, having integrated twice M1 Obtain answer 14 − 12 e− 1 , or exact equivalent A1 [5]

Question 13

0 3 Find the exact value of x2 sin 2x dx. [5] Ó 0

Accepted Answer

3 Integrate by parts and reach ax 2 cos2 x + b ∫ x cos2 x dx M1* Obtain − 12 x 2 cos2 x +∫ x cos2 x , or equivalent A1 Complete the integration and obtain − 12 x 2 cos2 x + 12 x sin 2 x + 14 cos2 x , or equivalent A1 Use limits correctly having integrated twice DM1* Obtain answer 18 (π 2 − 4) , or exact equivalent, with no errors seen A1 [5] 2 2ln x

Question 14

(i) Prove the identity tan tan sec [4] 21 −tan 1  1 21. 160 1 3 (ii) Hence show that tan sec ln [4] 2 2. Ó 0 1 21 d1 =

Accepted Answer

5 (i) EITHER: Use tan 2A formula to express LHS in terms of tanθ M1 Express as a single fraction in any correct form A1 Use Pythagoras or cos 2A formula M1 Obtain the given result correctly A1 OR: Express LHS in terms of sin 2θ, cos 2θ, sin θand cos θ M1 Express as a single fraction in any correct form A1 Use Pythagoras or cos 2A formula or sin(A – B) formula M1 Obtain the given result correctly A1 [4] (ii) Integrate and obtain a term of the form a ln(cos2θ) or b ln(cosθ) (or secant equivalents) M1* Obtain integral − 12 ln(cos 2θ) + ln(cosθ) , or equivalent A1 Substitute limits correctly (expect to see use of both limits) DM1 Obtain the given answer following full and correct working A1 [4]

Question 15

y M x O 12x e The diagram shows a sketch of the curve y for x 0, and its minimum point M. x = > (i) Find the x-coordinate of M. [4] .......…

Accepted Answer

7(i) Use correct quotient rule or product rule M1 Obtain correct derivative in any form A1 Equate derivative to zero and solve for x M1 Obtain x = 2 A1 Total: 4 7(ii) State or imply ordinates 1.6487…, 1.3591…, 1.4938… B1 Use correct formula, or equivalent, with h = 1 and three ordinates M1 Obtain answer 2.93 only A1 Total: 3 7(iii) Explain why the estimate would be less than E B1 Total: 1

Question 16

603 Showing all necessary working, ﬁnd the value of x cos 3x dx, giving your answer in terms of Ó 0 0. [5] ................................…

Accepted Answer

3 Integrate by parts and reach ax sin3 x + b ∫ sin3 xdx M1* 1 `1 A1 Obtain x sin3 x − ∫ sin3 x d x , or equivalent 3 3 1 1 A1 Complete the integration and obtain x sin3 x + cos3 x , or equivalent 3 9 Substitute limits correctly having integrated twice and obtained ax sin 3x + b cos 3x M1(dep*) 1 A1 Obtain answer (π − 2 ) OE 18 Total: 5

Question 17

cos x 7 A curve has equation y for 2 sin x, = −120 ≤x ≤120. + (i) Find the exact coordinates of the stationary point of the curve. [6] ....…

Accepted Answer

7(i) Use correct quotient or product rule M1 Obtain correct derivative in any form A1 ( ) ( ) 2 d 3sin 2 sin 3cos cos d 2 sin − + − = + y x x x x x x Condone invisible brackets if recovery implied later. Equate numerator to zero M1 Use 2 2 cos sin 1 + = x x and solve for sin x M1 6sin 3 0 − − = x ⇒ sinx =…. Obtain coordinates / 6 π = − x and 3 = y ISW A1 + A1 From correct working. No others in range SR: A candidate who only states the numerator of the derivative, but justifies this, can have full marks. Otherwise they score M0A0M1M1A0A0 6 7(ii) State indefinite integral of the form k ln (2 + sin x) M1* Substitute limits correctly, equate result to 1 and obtain 3 ln (2 + sin a) – 3 ln 2 = 1 A1 or equivalent Use correct method to solve for a M1(dep*) Allow for a correct method to solve an incorrect equation, so long as that equation has a solution. 1 3 1 2 1 sin e + = a ( ) 1 3 1 sin 2 e 1 − ⇒ = −     a Can be implied by 52.3° Obtain answer a = 0.913 or better A1 Ignore additional solutions. Must be in radians. 4

Question 18

π 4 Find x sec2x dx. Give your answer in a simpliﬁed exact form. [7] Ó 1 6π ...............................................................…

Accepted Answer

4 Integrate by parts and reach tan tan d +  ax x b x x Obtain tan tan d − x x x x A1 Complete the integration, obtaining a term lncos ± x , or equivalent M1 Obtain integral tan lncos + x x x, or equivalent A1 Substitute limits correctly, having integrated twice DM1 Use a law of logarithms M1 Obtain answer 5 1 3 ln3 18 2 π − , or exact simplified equivalent A1 7

Question 19

y M x O 1 x The diagram shows the curve y , for x and its maximum point M. = 1 3x4 ≥0, + (a) Find the x-coordinate of M, giving your answer…

Accepted Answer

6(a) Use quotient or product rule M1 Obtain correct derivative in any form e.g. ( ) ( ) 4 3 2 4 1 3 12 1 3 x x x x + − × + A1 Equate derivative to zero and solve for x M1 Obtain answer 0.577 A1 4 Question Answer Marks 6(b) State or imply d 2 3 d = u x x, or equivalent B1 Substitute for x and dx M1 Obtain integrand ( ) 2 1 2 3 1+ u , or equivalent A1 State integral of the form ܽtanିଵݑ and use limits u = 0 and u = 3 (or x = 0 and x = 1) correctly M1 Obtain answer 3 π 18 , or exact equivalent A1 5

Question 20

y M x O The diagram shows the curve y x 5 3 and its maximum point M. = + −2x (a) Find the exact coordinates of M. [5] .....................…

Accepted Answer

10(a) Use the product rule correctly to obtain 1 2 ( 5)(3 2 ) (3 2 ) n p x x q x     BOD over sign errors unless an incorrect rule is quoted. Obtain correct derivative in any form A1 e.g. 1 1 2 2 ( 5)(3 2 ) (3 2 ) x x x       . Equate derivative to zero and obtain a linear equation DM1 Allow with surd factor e.g.        1 2 3 2 5 3 2 0 x x x        . Obtain a correct linear equation. A1 e.g. –(x + 5) + 3 – 2x = 0. Obtain answer 2 13 39 , 3 9          . A1 Or exact equivalent e.g. 2 13 13 , 3 3 3          or 2 2197 , 3 27          . Accept with x, y stated separately. ISW Alternative Method for Question 10(a) Obtain y2 and differentiate *M1 Ignore their left hand side i.e. their 2 d d y x . Obtain correct derivative in any form A1 e.g. 2 6 34 20 x x    . Equate derivative to zero and solve for x DM1 Obtain 2 3  A1 Ignore –5 if seen. Obtain answer 2 13 39 , 3 9          only A1 Or exact equivalent e.g. 2 13 13 , 3 3 3          or 2 2197 , 3 27          . ISW 5 Question Answer Marks Guidance 10(b) Use the given substitution and reach 1 2 13 d 2 2 u a u u         *M1 OE Need to see -2 or -½ used. Condone if du missing or the integral sign is missing. Allow M1A0 for complete substitution into 3 2 d x x x   to obtain first term of the line below. Obtain correct integral 1 2 1 13 d 2 2 2 u u u          A1 OE e.g. 1 3 d 5 d 2 2 u u u u u            . Ignore limits at this stage. Condone if du missing. x = –5 and 3 2 B1 SOI e.g. by u = 13 and 0. In any order and at any stage. Use correct limits the right way round in an integral of the form 3 5 2 2 26 2 3 5 a u u          DM1 Obtain answer 169 13 15 or a = 169 15 A1 or exact equivalents. 5

Question 21

(a) Show that cos 4 i - sin 4 i / cos2i . [3] .............................................................................................…

Accepted Answer

7(a) Factorise LHS using difference of 2 squares *M1 2 2 2 2 cos sin cos sin       Simplify DM1 2 2 cos sin 1     must be seen or implied, e.g.     2 2 2 2 2 2 cos sin cos sin cos sin .           Obtain 4 4 cos sin cos2      from correct working A1 AG Alternative Method for Question 7(a) Use of correct rearrangements of double angle formulae (*M1) E.g. 2 2 1 cos2 1 cos2 2 2                  Only condone 2 2 1 cos2 cos2 1 2 2                  if correct expression for 2 sin  seen. Expand and simplify (DM1) Collect like terms. Condone recovery from missing brackets. Obtain 4 4 cos sin cos2      from correct working (A1) AG Alternative Method 2 for Question 7(a) Correct use of Pythagoras (*M1) E.g.   2 2 4 1 sin sin     or     2 2 2 2 cos 1 sin sin 1 cos        Expand and simplify (DM1) Condone recovery from missing brackets. Obtain 4 4 cos sin cos2      from correct working (A1) AG 3 Question Answer Marks Guidance 7(b) Use part (a) and correct double angle formula to obtain expression involving 2 sin 2 d   or 2 cos 2 d   M1 4 4 2 2 2 cos sin 4sin cos d cos2 sin 2 d             Allow BOD for 2 2sin 2if sin2 2sin cos     seen. 1 2 cos2 sin 2 d     B1 Seen or implied. Use of correct double angle formula on second part of the integral to obtain a form that can be integrated directly M1 e.g. 2 1 cos4 sin 2 d d 2       Obtain 1 1 2 8 sin4    A1 Condone a mixture of x and .  Correct use of limits π 8  in an expression of the form sin 2 sin 4 p q r      and evaluate the trig M1     1 1 1 2 16 8 2 2     Obtain 1 1 1 2 8 4 2    A1 ISW Or exact equivalent from exact working. 6

Question 22

- 7x 2 + 2 x - 6 10 Let f ( x) = . `1 + x`j4 + x 2j (a) Express f ( )x in partial fractions. [5] ..........................................…

Accepted Answer

10(a) A Bx + C B1 State or imply the form + 4 + x 2 (1 + x ) ( ) Use a correct method for finding a constant Even with incorrect PF denominators M1 A( 4 + x 2 ) + (Bx + C)(1 + x) + = −7 x 2 +2x − 6 Obtain one of A = –3, B = – 4 and C = 6 A1 Obtain a second value A1 Obtain a third value A1 A C Special Case 1: + + Find A, 4 + x 2 (1 + x ) ( ) M1 A1. Max 2/5. A Bx Special Case 2: + + Find A, 4 + x 2 (1 + x ) ( ) M1 A1. Max 2/5. 5 10(b) Obtain term –3ln (1 + x) B1 FT OE FT A ln (1 + x) Obtain term –2ln (4 + x2) B1 FT OE B FT ln (4 + x2) 2 Obtain integral of the form c tan−dx1 with d ≠ 1 following separation into two M1 d = 12 or 2 only. expressions −x1 A1 FT C −1 x Obtain 3tan FT tan 2 2 2 1 where Substitute correct limits correctly in an expression (obtained correctly) of the form M1 a ln (3) + b ln (8) − b ln (4) + c( 4 π ) , 1 where a, b, c ≠ 0. a ln (1 + x), b ln (4 + x2), and c tan −1 ( 2 x ) , a, b, c ≠ 0. 1 Do not allow slips, and must get to c ( 4 π ) . 3 A1 Must be in the form aπ – ln b. Obtain answer π − ln108 4 6

Question 23

2 3 Find the exact value of ; 1 3 cos 5 x d x . [4] r 5 ...................................................................................…

Accepted Answer

3 Use correct double angle formula M1* 1 2 (1 + cos10x ) Obtain 203 sin10x + 23 x A1 OE Use correct limits correctly DM1 3 15π 12π ( − 0 ) + − 20 40 40 Obtain 203 + 403 π A1 Or exact simplified equivalent. 4

Question 24

(a) Prove the identity sin 4x / 4 sin x `2 cos 3 x - cos xj. [3] ..........................................................................…

Accepted Answer

8(a) Use correct double angle formula to expand sin4x *M1 2sin2 x cos2 x OE OE. 2cos 2 x − 1 Use correct double angle formulae to obtain an expression in sin x and cos x dM1 E.g., 4sin x cos x ( ) 3 A1 AG Obtain sin4 x  4sin x 2cos x − cos x from fully correct working ( ) Alternative Method for Question 8(a) Use correct double angle formula to rearrange the RHS *M1 2 E.g. 2sin2 x 2cos x − 1 ( ) Use correct double angle formula to obtain an expression in sin2 x and cos2 x dM1 E.g. 2sin2 x cos2 x 3 A1 AG Obtain sin4 x  4sin x 2cos x − cos x from fully correct working ( ) Alternative Method 2 for Question 8(a) Use correct angle sum formulae to expand sin4x as far as an expression in *M1 E.g. sin x ( cos x cos2 x − sin x sin2 x ) sin x, cos x, sin2 x and cos2 x + cos x ( sin x cos2 x + cos x sin 2 x ) Use correct double angle formulae to obtain an expression in sin x and cos x dM1 2 3 E.g., sin x cos x 2cos x − 1 − 2sin x cos x ( ) + sin x cos x 2cos 2 x − 1 + 2sin x cos 3 x ( ) 3 A1 Obtain sin4 x  4sin x 2cos x − cos x from fully correct working ( ) 3 8(b) 6 4 B1 6 4 sin x cos x dx sin x cos x dx 8 Use the identity to obtain p  sin x cos x dx − 4   sin x cos x dx + q  Accept terms of the correct form but without the integral signs or the dx. Obtain r cos 7 x + s cos 5 x B1 Obtain − 8 cos 7 x + 4 cos 5 x B1 If using the substitution u = cos x, accept the form 7 5 8 7 4 5 − u + u . 7 5 Use the correct limits correctly in an expression of the form r cos 7 x + s cos 5 x M1 OE 8 1 4 1 8 4 Must be evaluated, e.g. −  +  + − . 7 8 2 5 4 2 7 5 1 A1 Or exact simplified equivalent (e.g. as 2 terms). Obtain 2 + 12 35 ( ) 5

Question 25

r 6 Find the exact value of x 2 sin 2 x dx . [6] 1y 0 .....................................................................................…

Accepted Answer

6 Commence integration by parts and reach Ax 2 cos2 x −∫ Bx cos2 xdx *M1 OE Condone sign error in formula. 1 2 A1 OE Obtain − x cos2 x + ∫ x cos2 xd x Unsimplified. 2 Complete integration by parts and reach Ax 2 cos2 x + Bx sin 2 x + C cos2 x *M1 OE ∫ Bx cos2 xd x = Bx sin 2 x + C cos2 x may be written separately to Ax 2 cos2 x for M1, but not for A1. 1 2 1 1 A1 OE Obtain − x cos2 x + x sin 2 x + cos2 x Unsimplified. 2 2 4 Substitute limits correctly in an expression of the form DM1  π 2 1   π   3   1  Ax 2 cos2 x + Bx sin 2 x + C cos2 x and evaluate the trig expressions A    + B     + C   − C  6  2   6   2   2  Allow one slip, including omitting the ‘– C’. Do not allow only decimals A1 ISW 1 2 3 1 Obtain answer − π + π − or exact three-term equivalent Allow equivalent fractions. 144 24 8 1 Allow 0.125 for . 8 6

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Integration — Paper 3 · A Level Mathematics 9709

Questions22 pages

Mark scheme25 answers

Another paper, or another topic

Questions as text

Q1 · 1 Find the exact value of the constant k for which dx = 1 9709/31 Oct/Nov 2007

Q2 · Y P x O T N 12 In the diagram the tangent to a curve at a general point P with… 9709/31 May/June 2008

Q3 · Use the substitution x 2 tan θ to show that = 2 1 8 4π dx cos2 θ dθ 9709/32 Oct/Nov 2009

Q4 · 2x 7 5 Show that dx ln 50 9709/33 Oct/Nov 2010

Q5 · 7 The integral I is deﬁned by I 4t3 dt 9709/31 May/June 2011

Q6 · 2x 23 Show that 4e −1 [5] dx = −2 9709/33 May/June 2011

Q7 · Y M 1 x O 2p The diagram shows the curve y 5 sin3x cos2x for 0 2π, and its maximum point M 9709/33 May/June 2011

Q8 · Use the substitution u tan x to show that, for n = ≠−1, 14π 1 dx ã 0 (tann+2x + tannx) =… 9709/33 Oct/Nov 2011

Q9 · 8 (a) Show that 4x ln x dx 56 ln 2 [5] Ó2 = −12 9709/31 May/June 2013

Q10 · Prove that cot tan 2 cosec [3] 1 + 1 21 9709/31 Oct/Nov 2013

Q11 · Prove that cot + tan 2 cosec 2 9709/32 Oct/Nov 2013

Q12 · 2 Find the exact value of dx 9709/31 May/June 2016

Q13 · 0 3 Find the exact value of x2 sin 2x dx 9709/32 May/June 2016

Q14 · Prove the identity tan tan sec [4] 21 −tan 1 1 21 9709/31 Oct/Nov 2016

Q15 · Y M x O 12x e The diagram shows a sketch of the curve y for x 0, and its minimum point M 9709/33 May/June 2017

Q16 · 603 Showing all necessary working, ﬁnd the value of x cos 3x dx, giving your answer in… 9709/33 May/June 2018

Q17 · Cos x 7 A curve has equation y for 2 sin x, = −120 ≤x ≤120 9709/32 Oct/Nov 2018

Q18 · Π 4 Find x sec2x dx 9709/32 Feb/March 2020

Q19 · Y M x O 1 x The diagram shows the curve y , for x and its maximum point M 9709/32 May/June 2020

Q20 · Y M x O The diagram shows the curve y x 5 3 and its maximum point M 9709/32 May/June 2023

Q21 · Show that cos 4 i - sin 4 i / cos2i 9709/32 May/June 2024

Q22 · - 7x 2 + 2 x - 6 10 Let f ( x) = 9709/32 Feb/March 2025

Q23 · 2 3 Find the exact value of ; 1 3 cos 5 x d x 9709/33 May/June 2025

Q24 · Prove the identity sin 4x / 4 sin x `2 cos 3 x - cos xj 9709/32 Oct/Nov 2025

Q25 · R 6 Find the exact value of x 2 sin 2 x dx 9709/33 Oct/Nov 2025