Pure: Integration Depth: Fluency and Exam Drill - Worksheets, Questions and Revision

20 original exam-style questions - 3 pages of questions with a full mark scheme - free printable PDF.

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A-Level · Pure Mathematics

P15D Pure: Integration Depth: Fluency and Exam Drill

EDEXCEL 9MA0 · Calculator allowed · about 50 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Find the integral of 8x(x²+3)³ dx.

(Total for Question 1 is 1 mark)

Find the integral of e^3x dx.

(Total for Question 2 is 1 mark)

Find the integral of 1/(2x+1) dx, for x > -1/2.

(Total for Question 3 is 1 mark)

Find the integral of cos(4x) dx.

(Total for Question 4 is 1 mark)

Find the integral of 6x²(x³-1)² dx.

(Total for Question 5 is 1 mark)

Find the integral of 1/x dx, for x > 0.

(Total for Question 6 is 1 mark)

Find the integral of sec²(x) dx.

(Total for Question 7 is 1 mark)

Find the exact value of the definite integral, from x = 0 to x = 2, of 3x² dx.

(Total for Question 8 is 1 mark)

Find the integral of 5e^2x dx.

(Total for Question 9 is 1 mark)

Find the integral of 4/(x+2) dx, for x > -2.

(Total for Question 10 is 1 mark)

Using the substitution u = 3 + x², find the integral of x √3+x² dx.

(Total for Question 11 is 2 marks)

A curve satisfies dy/dx = 6x²/y, where y > 0, and passes through the point (0, 3). By separating the variables, find y in terms of x.

(Total for Question 12 is 2 marks)

Using the substitution u = sin(x), find the integral of sin²(x)cos(x) dx.

(Total for Question 13 is 2 marks)

Find the exact value of the definite integral, from x = 1 to x = e, of 1/x dx.

(Total for Question 14 is 2 marks)

f(x) = x e^3x.

(a)Find the integral of f(x) dx, using integration by parts.(2)

(b)Hence find the exact value of the definite integral, from x = 0 to x = 1, of f(x) dx.(2)

(Total for Question 15 is 4 marks)

Find the integral of sin²(x)cos³(x) dx.

(Total for Question 16 is 3 marks)

The curve C has equation y = 1/√16-x², for -4 < x < 4.

(a)Find the integral of 1/√16-x² dx.(2)

(b)Hence show that the definite integral, from x = 0 to x = 2, of 1/√16-x² dx, equals π/6.(2)

(Total for Question 17 is 4 marks)

f(x) = (5x - 7) / ((x-3)(x+1)).

(a)Express f(x) in the form A/(x-3) + B/(x+1), where A and B are constants to be found.(2)

(b)Hence find the integral of f(x) dx.(2)

(Total for Question 18 is 4 marks)

The curve C has equation y = 6x - x². The line l has equation y = 2x. Find the area of the finite region bounded by C and l.

(Total for Question 19 is 3 marks)

The number of bacteria, N, in a culture is modelled by the differential equation dN/dt = N/50, where t is measured in hours. Initially there are 200 bacteria.

(a)By separating the variables, show that N = 200 e^t/50.(2)

(b)Find the time taken, in hours to 3 significant figures, for the population to reach 500.(2)

(Total for Question 20 is 4 marks)

Mark scheme · P15D Pure: Integration Depth: Fluency and Exam Drill

Question 1

B1 (x²+3)⁴ + c cao
Answer: (x²+3)⁴ + c

Question 2

B1 (1/3)e^3x + c cao
Answer: (1/3)e^3x + c

Question 3

B1 (1/2)ln(2x+1) + c cao
Answer: (1/2)ln(2x+1) + c

Question 4

B1 (1/4)sin(4x) + c cao
Answer: (1/4)sin(4x) + c

Question 5

B1 (2/3)(x³-1)³ + c cao
Answer: (2/3)(x³-1)³ + c

Question 6

B1 ln(x) + c cao
Answer: ln(x) + c

Question 7

B1 tan(x) + c cao
Answer: tan(x) + c

Question 8

B1 8 cao
Answer: 8

Question 9

B1 (5/2)e^2x + c cao
Answer: (5/2)e^2x + c

Question 10

B1 4ln(x+2) + c cao
Answer: 4ln(x+2) + c

Question 11

M1 substitutes u = 3+x², du = 2x dx, to get integral of (1/2)√u du
A1 = (1/3)(3+x²)^3/2 + c cao
Answer: (1/3)(3+x²)^3/2 + c

Question 12

M1 separates variables and integrates: y²/2 = 2x³ + C
A1 uses (0,3) to find C = 4.5, giving y = √4x³+9
Answer: y = √4x³ + 9

Question 13

M1 substitutes u = sin(x), du = cos(x) dx, to get integral of u² du
A1 = sin³(x)/3 + c cao
Answer: sin³(x)/3 + c

Question 14

M1 integrates to [ln x] between 1 and e
A1 = ln(e) - ln(1) = 1 cao
Answer: 1

Question 15

(a) M1 applies integration by parts with u=x, dv=e^3xdx
(a) A1 (x/3)e^3x - (1/9)e^3x + c
(a) Answer: (x/3)e^3x - (1/9)e^3x + c
(b) M1 substitutes the limits x=0 and x=1 into the result of part (a)
(b) A1 (2e³ + 1)/9 cao (exact)
(b) Answer: (2e³ + 1)/9

Question 16

M1 rewrites cos³(x) = cos(x)(1-sin²(x)) and substitutes u = sin(x)
A1 integrates to u³/3 - u⁵/5
A1 = sin³(x)/3 - sin⁵(x)/5 + c cao
Answer: sin³(x)/3 - sin⁵(x)/5 + c

Question 17

(a) M1 uses the standard result for integral of 1/√a²-x² dx with a=4
(a) A1 arcsin(x/4) + c cao
(a) Answer: arcsin(x/4) + c
(b) M1 substitutes the limits into the result of part (a): arcsin(0.5) - arcsin(0)
(b) A1 cso: = π/6, as required
(b) Answer: π/6

Question 18

(a) M1 sets up A(x+1) + B(x-3) = 5x-7 and compares coefficients (or substitutes x=3, x=-1)
(a) A1 A = 2, B = 3
(a) Answer: A = 2, B = 3
(b) M1 integrates each partial fraction term (ft from (a))
(b) A1 2ln(x-3) + 3ln(x+1) + c cao
(b) Answer: 2ln(x-3) + 3ln(x+1) + c

Question 19

M1 finds the intersections x=0 and x=4 by solving 6x-x²=2x
M1 sets up the integral of (4x-x²) dx between 0 and 4
A1 area = 32/3 cao
Answer: 32/3

Question 20

(a) M1 separates variables and integrates: ln N = t/50 + C
(a) A1 cso: uses N=200 at t=0 to obtain N = 200e^t/50
(a) Answer: N = 200e^t/50
(b) M1 sets 200e^t/50 = 500 and takes logarithms of both sides
(b) A1 awrt 45.8 hours
(b) Answer: awrt 45.8 hours

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Pure: Integration Depth: Fluency and Exam Drill - Worksheets, Questions and Revision

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P15D Pure: Integration Depth: Fluency and Exam Drill

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Question 20

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