ks5 full coverage: integration (year 2)
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KS5 "Full Coverage": Integration (Year 2)
This worksheet is designed to cover one question of each type seen in past papers, for each A
Level topic. This worksheet was automatically generated by the DrFrostMaths Homework
Platform: students can practice this set of questions interactively by going to
www.drfrostmaths.com, logging on, Practise β Past Papers (or Library β Past Papers for
teachers), and using the βRevisionβ tab.
Question 1 Categorisation: Integrate exponential terms.
[OCR C3 June 2011 Q1i] Find
β« 6π2π₯+1ππ₯
.......................... +π
Question 2 Categorisation: In general, integrate expressions of the form πβ²(ππ + π)
Determine the following, using π as your constant of integration.
β« sin(3π₯) ππ₯
..........................
Question 3 Categorisation: Appreciate the need to expand and simplify to integrate some
expressions.
[OCR C3 June 2016 Q2i] Find
β« (2 β1
π₯)
2
ππ₯
.......................... +π
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Question 4 Categorisation: Split fractions in order to integrate.
[Edexcel A2 SAM P1 Q4] Given that a is a positive constant and
β«π‘ + 1
π‘ππ‘ = ππ 7
2π
π
show that π = ππ π , where π is a constant to be found.
..........................
Question 5 Categorisation: Integrate expressions of the form (ππ + π)π where π is negative or
fractional.
[Edexcel C4 June 2016 Q7a] Find
β«(2π₯ β 1)32ππ₯
giving your answer in its simplest form.
.......................... +π
Question 6
Categorisation: Rewrite expressions in the form π
(ππ+π)π as (ππ + π)βπ in order to
integrate.
Find
β«3
β5π₯ β 1 ππ₯
.......................... +π
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Question 7 Categorisation: Integrate common trig expressions.
Determine β« π‘ππ 2π₯ππ₯
.......................... +π
Question 8 Categorisation: As above.
Determine β« π ππ 23π₯ππ₯
.......................... +π
Question 9 Categorisation: As above.
Determine β« πππ ππ2π₯ππ₯
.......................... +π
Question 10 Categorisation: Integrate πππππ and πππππ.
Determine β« π ππ 2π₯ππ₯
.......................... +π
Question 11 Categorisation: Integrate ππ for some constant π.
Determine β« 3π₯ππ₯
.......................... +π
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Question 12 Categorisation: Integrate by inspection.
Integrate the following, using π as your constant of integration.
β« π₯πβπ₯2ππ₯
..........................
Question 13
Categorisation: Integrate by inspection, appreciating that β«πβ²(π)
π(π)π π = ππ π(π) + π
Determine β«π₯
π₯2+1ππ₯
.......................... +π
Question 14 Categorisation: Integrate by splitting into partial fractions.
[Edexcel C4 June 2016 Q6i]
Given that π¦ > 0 , find
β«3π¦ β 4
π¦(3π¦ + 2)ππ¦
.......................... +π
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Question 15
Categorisation: Integrate by inspection for expressions of the form π
(ππ+π)π
[Edexcel C4 June 2014 Q6ii] Find
β«8
(2π₯ β 1)3ππ₯
where π₯ >1
2 .
.......................... +π
Question 16
Categorisation: Appreciate the difference in techniques between integrating π
π and
integrating π
ππ
[Edexcel C4 June 2012 Q1bi Edited]
π(π₯) =1
π₯(3π₯ β 1)2=
1
π₯β
3
3π₯ β 1+
3
(3π₯ β 1)2
Find β« π(π₯)ππ₯ .
.......................... +π
Question 17 Categorisation: Integrate by parts.
[Edexcel C4 June 2014 Q6i] Find
β« π₯π4π₯ππ₯
β« π₯π4π₯ππ₯ = .......................... +π
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Question 18 Categorisation: Definite integration by parts.
[Edexcel C4 Jan 2011 Q1] Use integration to find the exact value of
β« π₯ π ππ 2π₯ππ₯
π2
0
..........................
Question 19 Categorisation: Definite integration by parts to find an area.
[Edexcel A2 Specimen Papers P1 Q7]
Figure 4 shows a sketch of part of the curve with equation π¦ = 2π2π₯ β π₯π2π₯ , π₯ β πππ‘βππ π
The finite region π , shown shaded in Figure 4, is bounded by the curve, the π₯ βaxis and the π¦
βaxis.
Use calculus to show that the exact area of π can be written in the form ππ4 + π , where π
and π are rational constants to be found.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
..........................
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Question 20 Categorisation: Integrate ππ π
Integrate the following, using π as your constant of integration.
β« ππ (π₯)ππ₯
..........................
Question 21 Categorisation: Integrate by parts involving ππ π.
[Edexcel C4 June 2012 Q7b]
Figure 3 shows a sketch of part of the curve with equation π¦ = π₯1
2 ππ 2π₯ .
The finite region R, shown shaded in Figure 3, is bounded by the curve, the π₯ -axis and
the lines π₯ = 1 and π₯ = 4 .
Find β« π₯1
2 ππ 2π₯ππ₯
.......................... +π
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Question 22 Categorisation: As above.
[Edexcel A2 SAM P1 Q14c]
Figure 4 shows a sketch of part of the curve πΆ with equation
π¦ =π₯2 ππ π₯
3β 2π₯ + 5 , π₯ > 0
The finite region π , shown shaded in Figure 4, is bounded by the curve πΆ , the line with
equation π₯ = 1 , the π₯ -axis and the line with equation π₯ = 3
Show that the exact area of π can be written in the form π
π+ ππ π , where π , π and π are
integers to be found.
π = ..........................
Question 23 Categorisation: Integrate by parts twice.
[Edexcel C4 June 2013 Q1a]
Find β« π₯2ππ₯ππ₯
.......................... +π
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Question 24 Categorisation: Integrate by parts where the technique would infinitely loop.
Determine β« ππ₯ π ππ π₯ππ₯
.......................... +π
Question 25 Categorisation: Integrate by substitution to determine an area.
[Edexcel C4 Jan 2013 Q4c]
Figure 1 shows a sketch of part of the curve with equation π¦ =π₯
1+βπ₯ . The finite region R,
shown shaded in Figure 1, is bounded by the curve, the π₯ -axis, the line with equation π₯ =
1 and the line with equation π₯ = 4 .
Use the substitution π’ = 1 + βπ₯ to find, by integrating, the exact area of R.
..........................
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Question 26 Categorisation: Integrate by substitution.
[Edexcel C4 June 2010 Q2 Edited]
Using the substitution π’ = πππ π₯ + 1 , or otherwise, to find the exact value of
β« π πππ π₯+1 π ππ π₯ππ₯
π2
0
..........................
Question 27 Categorisation: Definite integration by substitution when the substitution is not
given.
[Edexcel A2 Specimen Papers P1 Q12 Edited]
Show that
β« π ππ 2π
1 + πππ πππ = 2 + π ππ 2
π2
0
where π is a constant to be found.
..........................
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Question 28 Categorisation: As above.
[Edexcel C4 June 2016 Q6iia]
Use the substitution π₯ = 4 π ππ 2π to show that
β« βπ₯
4 β π₯ππ₯ = π β« π ππ 2πππ
π3
0
3
0
where π is a constant to be determined.
π = ..........................
Question 29 Categorisation: As above.
[Edexcel C4 Jan 2010 Q8a]
Using the substitution π₯ = 2 πππ π’ , or otherwise, find the exact value of
β«1
π₯2β4 β π₯2ππ₯
β2
1
..........................
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Question 30 Categorisation: As above.
[Edexcel C4 June 2013(R) Q3] Using the substitution π’ = 2 + β2π₯ + 1 , or other suitable
substitutions, find the exact value of
β«1
2 + β2π₯ + 1ππ₯
4
0
giving your answer in the form π΄ + 2 ππ π΅ , where π΄ is an integer and π΅ is a positive constant.
..........................
Question 31 Categorisation: Parametric integration.
[Edexcel A2 Specimen Papers P2 Q10b Edited]
Figure 4 shows a sketch of the curve πΆ with parametric equations
π₯ = ππ (π‘ + 2), π¦ =1
π‘ + 1, π‘ > β
2
3
The finite region π , shown shaded in Figure 4, is bounded by the curve πΆ , the line with
equation π₯ = ππ 2 , the π₯ -axis and the line with equation π₯ = ππ 4
Use calculus to find the exact area of π .
..........................
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Question 32 Categorisation: As above.
[Edexcel C4 Jan 2010 Q7b Edited]
π₯ = 5π‘2 β 4 , π¦ = π‘(9 β π‘2)
The curve C cuts the π₯ -axis at the points π΄ and π΅ . The π₯ -coordinate at the point π΄ is β4
and the π₯ -coordinate at the point π΅ is 41 .
The region R, as shown shaded in Figure 2, is enclosed by the loop of the curve.
Use integration to find the area of R.
..........................
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Question 33 Categorisation: Parametric integration involving an ππ term.
[Edexcel C4 Jan 2013 Q5d Edited]
Figure 2 shows a sketch of part of the curve C with parametric equations
π₯ = 1 β1
2π‘ , π¦ = 2π‘ β 1 .
The curve crosses the π¦ -axis at the point π΄ and crosses the π₯ -axis at the point π΅ .
The point π΄ has coordinates (0,3) and the point π΅ has coordinates (1,0) .
The region R, as shown shaded in Figure 2, is bounded by the curve C, the line π₯ = β1
and the π₯ βaxis.
Use integration to find the exact area of R.
..........................
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Question 34 Categorisation: Solve a differential equation.
[Edexcel C4 Jan 2010 Q5b Edited]
It is given that β«9π₯+6
π₯ππ₯ = 9π₯ + 6 ππ π₯ + π , π₯ > 0 .
Given that π¦ = 8 at π₯ = 1 , solve the differential equation
ππ¦
ππ₯=
(9π₯ + 6)π¦13
π₯
giving your answer in the form π¦2 = π(π₯) .
π¦2 = ..........................
Question 35 Categorisation: As above, but using trig functions.
[Edexcel C4 June 2014 Q6iii]
Given that π¦ =π
6 at π₯ = 0 , solve the differential equation
ππ¦
ππ₯= ππ₯ πππ ππ 2π¦ πππ ππ π¦
..........................
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Question 36 Categorisation: As above.
[Edexcel C4 June 2012 Q4] Given that π¦ = 2 at π₯ =π
4 , solve the differential equation
ππ¦
ππ₯=
3
π¦ πππ 2π₯
..........................
Question 37 Categorisation: As above.
[Edexcel C4 June 2013 Q6a Edited]
Water is being heated in a kettle. At time π‘ seconds, the temperature of the water is π
The rate of increase of the temperature of the water at any time π‘ is modelled by the
differential equation
ππ
ππ‘= π(120 β π) , π β€ 100
where π is a positive constant.
Given that π = 20 when π‘ = 0 , solve this differential equation to show that
π = π β ππβππ‘
where π and π are integers to be determined.
..........................
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Question 38 Categorisation: Answer questions about meerkats.
[Edexcel A2 SAM P2 Q16c Edited] It can be shown that
1
π(11 β 2π)β‘
1
11π+
2
11(11 β 2π)
A population of meerkats is being studied. The population is modelled by the
differential equation
ππ
ππ‘=
1
22π(11 β 2π) , π‘ β₯ 0 , 0 < π < 5.5
where π , in thousands, is the population of meerkats and π‘ is the time measured in years
since the study began. Given that there were 1000 meerkats in the population when the study
began, show that
π =π΄
π΅ + πΆπβ12
π‘
where π΄ , π΅ and πΆ are integers to be found.
..........................
Question 39 Categorisation: Understand the meaning of constants within a differential equation.
[Edexcel C4 June 2013(R) Q8a] In an experiment testing solid rocket fuel, some fuel is
burned and the waste products are collected. Throughout the experiment the sum of the
masses of the unburned fuel and waste products remains constant.
Let π₯ be the mass of waste products, in kg, at time π‘ minutes after the start of the
experiment. It is known that at time π‘ minutes, the rate of increase of the mass of waste
products, in kg per minute, is π times the mass of unburned fuel remaining, where π is a
positive constant.
The differential equation connecting π₯ and π‘ may be written in the form ππ₯
ππ‘= π(π β π₯) , where π is a constant.
Explain, in the context of the problem, what ππ₯
ππ‘ and π represent.
..........................
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Question 40 Categorisation: Integrate by parts requiring prior factorisation of the denominator.
[MAT 2003 1D] What is the exact value of the definite integral
β«ππ₯
π₯ + π₯3
2
1
..........................
Question 41 Categorisation: Use the double angle rule for sin to integrate.
Determine β« π ππ π₯ πππ π₯ππ₯
.......................... +π
Question 42 Categorisation: As above.
Determine π ππ 2π₯ πππ 2π₯ππ₯
.......................... +π
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Question 43 Categorisation: Integrate by inspection involving a power of a trig function.
Determine β« π ππ 3π₯ π‘ππ π₯ππ₯
.......................... +π
Question 44 Categorisation: As above.
Determine β« π ππ 3π₯ πππ π₯ππ₯
.......................... +π
Question 45 Categorisation: Use the trapezium rule to approximate the area under a curve.
[Edexcel A2 Specimen Papers P1 Q1b Edited]
Figure 1 shows a sketch of the curve with equation π¦ =π₯
1+βπ₯ , π₯ β₯ 0
The finite region π , shown shaded in Figure 1, is bounded by the curve, the line with equation
π₯ = 1 , the π₯ βaxis and the line with equation π₯ = 3
The table below shows corresponding values of π₯ and π¦ for π¦ =π₯
1+βπ₯
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The trapezium rule can be used to find an estimate for the area of π using the values in the
table.
Explain how the trapezium rule can be used to give a better approximation for the area of π .
..........................
Question 46 Categorisation: Use an area approximated using the trapezium rule to find a similar
area.
[Edexcel A2 Specimen Papers P1 Q1ci Edited] (Continued from above)
The finite region π , shown shaded in Figure 1, is bounded by the curve, the line with equation
π₯ = 1 , the π₯ βaxis and the line with equation π₯ = 3
The table below shows corresponding values of π₯ and π¦ for π¦ =π₯
1+βπ₯
Using the trapezium rule, with all the values of π¦ in the table, an estimate for the area of π is
given as 1.635
Giving your answer to 3 decimal places in each case, use this answer to deduce an estimate for
β«5π₯
1+βπ₯ππ₯
3
1
..........................
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Question 47 Categorisation: As above.
[Edexcel A2 Specimen Papers P1 Q1cii Edited] (Continued from above)
Figure 1 shows a sketch of the curve with equation π¦ =π₯
1+βπ₯ , π₯ β₯ 0
The finite region π , shown shaded in Figure 1, is bounded by the curve, the line with equation
π₯ = 1 , the π₯ βaxis and the line with equation π₯ = 3
The table below shows corresponding values of π₯ and π¦ for π¦ =π₯
1+βπ₯
Using the trapezium rule, with all the values of π¦ in the table, an estimate for the area of π is
given as 1.635
Giving your answer to 3 decimal places in each case, use this answer to deduce an estimate for
β« (6 +π₯
1+βπ₯) ππ₯
3
1
..........................
Question 48 Categorisation: Use the trapezium rule where the step size π needs to be
determined.
[OCR C2 June 2016 Q8v] Use the trapezium rule, with 2 strips each of width 1.5, to find an
estimate for β« 3π₯β2ππ₯4
1 .
Give your answer correct to 3 significant figures.
..........................
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Answers
Question 1
3π2π₯+1
Question 2
β1
3cos 3π₯ + π
Question 3
4π₯ β 4 ππ |π₯| β1
π₯
Question 4
π =7
2
Question 5
1
5(2π₯ β 1)
5
2
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Question 6
6
5β5π₯ β 1
Question 7
π‘ππ π₯ β π₯
Question 8
1
3 π‘ππ 3π₯
Question 9
β πππ‘ π₯
Question 10
1
2π₯ β
1
4 π ππ 2π₯
Question 11
3π₯
ππ 3
Question 12
β1
2πβπ₯2
+ π
Question 13
1
2 ππ |π₯2 + 1|
Question 14
β2 ππ π¦ + 3 ππ (3π¦ + 2)
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Question 15
β«8
(2π₯β1)3 ππ₯ = β2(2π₯ β 1)β2
Question 16
ππ π₯ β ππ (3π₯ β 1) β1
3π₯β1
Question 17
β« π₯π4π₯ππ₯ =1
4π₯π4π₯ β
1
16π4π₯
Question 18
π
4
Question 19
π =1
4 , π = β
5
4
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Question 20
π₯ ππ π₯ β π₯ + π
Question 21
2
3π₯
3
2 ππ 2π₯ β4
9π₯
3
2
Question 22
π =28
27+ ππ 27
Question 23
π₯2ππ₯ β 2(π₯ππ₯ β ππ₯)
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Question 24
1
2ππ₯( π ππ π₯ β πππ π₯)
Question 25
11
3+ 2 ππ 2 β 2 ππ 3
Question 26
π(π β 1)
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Question 27
π = β2
Question 28
π = 8
Question 29
β3β1
4
Question 30
2 + 2 ππ (3
5)
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Question 31
ππ (3
2)
Question 32
648
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Question 33
15
2 ππ 2β 2
Question 34
π¦2 = (6π₯ + 4 ππ π₯ β 2)3
Question 35
2
3 π ππ 3π¦ = ππ₯ β
11
12
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Question 36
1
2π¦2 = 3 π‘ππ π₯ β 1
Question 37
π = 120 , π = 100
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Question 38
π΄ = 11 , π΅ = 2 , πΆ = 9
Question 39
"rate of increase OR rate of change" and "mass"
Question 40
1
2 ππ
8
5
Question 41
β1
4 πππ 2π₯
π ππ π₯ πππ π₯ can be written as 1
2 π ππ 2π₯ .
Question 42
1
8π₯ β
1
32 π ππ 4π₯
π ππ 2π₯ πππ 2π₯ = ( π ππ π₯ πππ π₯)2
= (1
2 π ππ 2π₯)
2
=1
4 π ππ 22π₯
=1
8β
1
8 πππ 4π₯
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Question 43
1
3 π ππ 3π₯
Question 44
1
4 π ππ 4π₯
Question 45
"more OR increase OR decrease the width OR narrower OR smaller"
Question 46
8.175
Question 47
13.635
Question 48
9.6