Integration — Paper 2 · A Level Mathematics 9709

6 1 2 Show that dx 2 ln 2. [4] x 2 = ä 0 +

4 marks

Mark scheme: 2 Obtain integral ln(x + 2) B1 Substitute correct limits correctly M1 Use law for the logarithm of a product, a quotient or a power M1 Obtain given answer following full and correct working A1 [4]

This question in 9709/21 May/June 2010

4 Find the exact value of the positive constant k for which k 2k e4x dx ex dx. ã 0 = ã 0 [6]

6 marks

Mark scheme: 4 State at least one correct integral B1 Use limits correctly to obtain an equation in e2k, e4k M1 Carry out recognizable solution method for quadratic in e2k M1 Obtain e2k = 1 and e2k = 3 A1 Use logarithmic method to solve an equation of the form eλa = b, where b > 0 M1 1 Obtain answer k = ln 3 A1 [6] 2 1

This question in 9709/22 Oct/Nov 2011

6 (a) Use the trapezium rule with two intervals to estimate the value of 1 1 dx, 6 2ex ä 0 + giving your answer correct to 2 decimal places. [3] (b) Find dx. [4] (ex −2)2 ä e2x

7 marks

Mark scheme: 6 (a) State or imply correct ordinates 0.125, 0.08743…, 0.21511… B1 Use correct formula, or equivalent, correctly with h = 0.5 and three ordinates M1 Obtain answer 0.11 with no errors seen A1 [3] (b) Attempt to expand brackets and divide by e2x M1 Integrate a term of form ke−x or ke−2x correctly A1 Obtain 2 correct terms A1 Fully correct integral x + 4e−x - 2e−2x + c A1 [4]

This question in 9709/21 Oct/Nov 2012

16 6 6 (a) Show that dx ln 125. [5] 2x = Ô6 −7 (b) Use the trapezium rule with four intervals to find an approximation to 17 log10 x dx, Ó 1 giving your answer correct to 3 significant figures. [3]

8 marks

Mark scheme: 6 (a) Integrate to obtain form k ln( 2 x − 7 ) M1 Obtain correct 3 ln( 2 x − 7 ) A1 Substitute limits correctly (dependent on first M1) DM1 Use law for logarithm of a quotient or power (dependent on first M1) DM1 Confirm ln 125 following correct work and sufficient detail (AG) A1 [5] (b) Evaluate y at (1), 5, 9, 13, 17 M1 Use correct formula, or equivalent, with h = 4 and five y-values M1 Obtain 13.5 A1 [3] d

This question in 9709/21 May/June 2014

1 @ A 30 1 1 97 (i) Show that the exact value of cos2x dx is [6] cos2x 60 + 8ï3. Ô 0 + (ii) y x O 130 1 The diagram shows the curve y cos x for 0 The shaded region is bounded cosx = 1 + ≤x ≤130. by the curve and the lines x 0, x and y 0. Find the exact volume of the solid obtained when the shaded region is rotated= completely= 30 about= the x-axis. [4]

10 marks

Mark scheme: 7 (i) Express cos 2 x in form k1 + k 2 cos 2 x M1 Obtain correct 12 + 12 cos 2 x A1 Rewrite second term as sec 2 x B1 Integrate to obtain at least terms k 3 sin 2 x and k 4 tan x M1 Obtain 12 x + 14 sin 2 x + tan x A1 Confirm given result 16 π + 89 3 A1 [6] 1 (ii) State volume is π ∫ (cos x + cos x 2) (π maybe implied by later appearance) B1 1 1 + 2) dx B1 Expand to obtain π ∫ (cos 2 x + cos + 2) dx or ∫ (cos 2 x + cos 2 x 2 x Integrate integrand involving three terms (in part using part (i) or otherwise i.e. k 3 sin 2 x + k 4 tan x + k 5 x ) M1 Obtain 56 π 2 + 89 3π or exact equivalent A1 [4]

This question in 9709/22 Oct/Nov 2015

35 3 1 Find the exact value of dx, giving the answer in the form ln k. [5] Ô 2x 5 −1 +

5 marks

Mark scheme: 1 Integrate to obtain k ln( 2 x + 5) M1 Obtain correct 32 ln( 2 x + 5) A1 Apply limits and use logarithm law for ln a − ln b M1 Use logarithm power law M1 Obtain ln 125 A1 [5] 2 2

This question in 9709/23 Oct/Nov 2015

a 5 Given that dx 65, find the value of a correct to 3 decimal places. [5] = Ó0 6e2x+1

5 marks

Mark scheme: 5 Obtain integral of form ke 2 x +1 M1 Obtain correct 3e 2 x +1 A1 Apply both limits correctly and rearrange at least to e 2 a +1 = ... M1 Use logarithms correctly to find a M1 Obtain 1.097 A1 [5]

This question in 9709/22 Feb/March 2016

7 6 2 Show dx ln 125. [5] 2x 1 = thatÔ1 + … … … … … … … … … … … … … … … … … … … … … … … …

5 marks

Mark scheme: 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

This question in 9709/23 Oct/Nov 2018

3a 2 7 3 It is given that dx ln 2. 2x = Ôa −5 Find the value of the positive constant a. [6] … … … … … … … … … … … … … … … … … … … … … … …

6 marks

Mark scheme: 3 Integrate to obtain ln(2 5) k x − *M1 For non-zero constant k Apply limits to obtain 7 2 ln(6 5) ln(2 5) ln a a − − − = A1 Apply subtraction law for logarithms *M1 OE Obtain equation 6 5 7 2 5 2 a a − = − A1 OE without logarithms Solve equation for a DM1 Obtain 25 2 a = A1 6

This question in 9709/22 Feb/March 2020

7 (a) Find the quotient when 9x3 1 is divided by 3x 2 , and show that the remainder is 9. −6x2 −20x + + [3] … … … … … … … … … … … … 6 9x3 1 (b) Hence find dx, giving the answer in the form a ln b where a and b are −6x2 −20x + 3x 2 + Ô1 + integers. [5] … … … … … … … … … … … … … … … … … (c) Find the exact root of the equation 9e9y 0. [4] −6e6y −20e3y −8 = … … … … … … … … … … … … … … … … …

12 marks

Mark scheme: 7(a) Carry out division at least as far as 2 3 + x kx M1 Obtain quotient 2 3 4 4 − − x x A1 Confirm remainder is 9 AG A1 3 7(b) Integrate to obtain at least 3 1k x and 2 ln(3 2) + k x terms *M1 Obtain 3 2 2 4 3ln(3 2) − − + + x x x x (FT from quotient in part (a)) A1FT Apply limits correctly DM1 Apply appropriate logarithm properties correctly M1 Obtain 125 ln64 + A1 5 7(c) State or imply 3 2 2 9 6 20 8 (3 2)(3 4 4) − − − = + − − x x x x x x (FT from quotient in part (a)) B1FT Attempt to solve cubic eqn to find positive value of x (or of 3e y ) M1 Use logarithms to solve equation of form 3e = y k where 0 > k M1 Obtain 1 ln 2 3 or exact equivalent A1 4

This question in 9709/21 May/June 2020

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

4 marks

Mark scheme: 1 Integrate to obtain 2 B1 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

This question in 9709/23 Oct/Nov 2021

3 y A B x O 2 The diagram shows the curve with 2x. The points on the curve with equation y x-coordinates 0 and 2 are denoted and =The6e−1shaded region is enclosed by the curve, the line by A B respectively. through A parallel to the x-axis and the line through B parallel to the y-axis. (a) Find the exact gradient of the curve at B. [2] … … … … … … … … … … … … … … … (b) Find the exact area of the shaded region. [3] … … … … … … … … … … … … … … … … … … … … … … … … …

5 marks

Mark scheme: 3(a) − 12 x M1 For any non-zero k except 6. Differentiate to obtain form ke 1 3 A1 Substitute x = 2 to obtain − 3e− or − e 2 3(b) 2 x B1 OE Integrate to obtain −12e − 1 x M1 For any non-zero k except 6 1 Use limits 0 and 2 correctly to an integral of the form ke −, 2 retaining − or equivalent perhaps involving integration of 6 − 6e 2 x . exactness 1 12 A1 Subtract from 12 to obtain final answer 12e− or e 3

This question in 9709/22 Oct/Nov 2023

1 d y 2 12 A curve passes through the point with coordinates b r, 5l and is such that = 4 sec b xl. 2 d x 2 Find the equation of the curve. [3] … … … … … … … … … … … … … … … … … … … … … … … … … …

3 marks

Mark scheme: 2 Integrate to obtain the form y = k tan 12 x *M1 No need for c yet. Substitute x = 12 π and y = 5 to determine value of c DM1 Obtain y = 8tan 12 x − 3 A1 3

This question in 9709/22 Feb/March 2025

11 8 1 Show that dx = ln a , where a is an integer to be found. [3] y 2 4x + 1 … … … … … … … … … … … … … … … … … … … … … … … … … … …

3 marks

Mark scheme: Question Answer Marks Guidance 1 Integrate to obtain 2ln(4 x + 1) B1 Apply limits correctly to k ln(4 x + 1) and use at least one relevant logarithm property M1 Obtain 2ln45 − 2ln9 = 2ln5 or equivalent, and conclude ln25 A1 3

This question in 9709/23 May/June 2025

11 8 1 Show that dx = ln a , where a is an integer to be found. [3] y 2 4x + 1 … … … … … … … … … … … … … … … … … … … … … … … … … … …

3 marks

Mark scheme: Question Answer Marks Guidance 1 Integrate to obtain 2ln(4 x + 1) B1 Apply limits correctly to k ln(4 x + 1) and use at least one relevant logarithm property M1 Obtain 2ln45 − 2ln9 = 2ln5 or equivalent, and conclude ln25 A1 3

This question in 9709/25 May/June 2025