ASMaths: PureOCRExam Questions8. Integration8.1 Integration

Integration (OCR AS Maths: Pure)

Exam Questions

4 hours41 questions
Easy
Medium
Hard
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1
Sme Calculator3 marks

Integrate

(i)
2x,
(ii)
6x2,
(iii)
1 half x to the power of begin inline style 1 half end style end exponent.

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2
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Evaluate

(i)
integral subscript 1 superscript 2 4 x space straight d x,

(ii)
integral subscript 0 superscript 3 open parentheses 9 x squared plus 4 x close parentheses space straight d x.

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3
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Use calculus to find

integral open parentheses 3 x to the power of 1 half end exponent plus 2 x to the power of negative 1 half space end exponent close parentheses straight d x.

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4a
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Show that

fraction numerator 3 x cubed plus 4 x to the power of 6 over denominator x squared end fraction

can be written as  3 x to the power of a plus 4 x to the power of b, where a and b are constants to be found.

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4b
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Hence find

integral open parentheses fraction numerator 3 x cubed plus 4 x to the power of 6 over denominator x squared end fraction close parentheses space straight d x.

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5a
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Integrate  5 x to the power of 4 plus 6 x squared plus 2 x plus 3.

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5b
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Given that space straight f left parenthesis x right parenthesis equals integral left parenthesis 5 x to the power of 4 plus 6 x squared plus 2 x plus 3 right parenthesis space d x and that the graph of space y equals straight f left parenthesis x right parenthesis  passes through the point (1 , 10), find an expression for straight f left parenthesis x right parenthesis in terms of x only.

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6
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Evaluate

integral subscript negative 3 end subscript superscript 4 open parentheses 2 k x plus 3 k x squared close parentheses space straight d x

giving your answer in terms of k.

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7
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The curve C, described by the integral

y equals integral open parentheses 2 x cubed minus x close parentheses space straight d x comma

passes through the point (2 , -2).

Show that space 2 y equals x to the power of 4 minus x squared minus 16.

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8a
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The area bounded by the curve with equation space y equals 9 minus x squared, the x-axis and the vertical lines with equations x equals 1 and x equals 2 is to be found.

Write down an integral that would find this area.

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8b
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Evaluate your integral from part (a) and hence find the area described above.

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9
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The diagram below shows the graph of y equals 7 x minus x squared minus 6.

q9-8-1-integration-easy-a-level-maths-pure-screenshot

Find the shaded area, giving your answer as a fraction in its simplest terms.

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10
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The diagram below shows the graph of space y equals x squared minus 8 x plus 12.

q10-8-1-integration-easy-a-level-maths-pure-screenshot

Find the shaded area marked R, giving your answer as a fraction in its simplest form.

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11a
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Simplify space x minus open parentheses x squared minus 8 x plus 18 close parentheses.

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11b
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The diagram below shows the graphs of  y equals x squared minus 8 x plus 18 space and  y equals x.

q11-8-1-integration-easy-a-level-maths-pure-screenshot

Find the shaded area marked R, giving your answer as a fraction in its simplest terms.

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This topic pack contains questions on the following:

  • Fundamental Theorem of Calculus

  • Integrating Powers of x

  • Definite Integration

  • Indefinite Integration

  • Area Under a Curve

  • Area between a Curve and a Line

1
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Use calculus to find

space space integral open parentheses 3 x squared plus 5 x plus 3 close parentheses space straight d x

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2
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Evaluate

space integral subscript 1 superscript 5 left parenthesis 4 x plus 6 x squared right parenthesis space straight d x

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3
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Find the equation of the curve passing through the point (4, 64) and given by

  y equals integral open parentheses 3 minus 2 x plus 6 x squared close parentheses space straight d x

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4a
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Show that

open parentheses 3 minus 2 x close parentheses squared equals 9 minus 12 x plus 4 x squared space

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4b
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Hence, or otherwise, work out

integral open parentheses 3 minus 2 x close parentheses squared space straight d x

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5
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Given

integral subscript k superscript 5 space space left parenthesis 2 x minus 1 right parenthesis space straight d x equals 20

find the value of the positive constant k.

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6a
Sme Calculator1 mark

The diagram below shows part of the graph ofspace y equals 4 minus open parentheses x minus 2 close parentheses squared.

q5-8-1-integration-easy-a-level-maths-pure-screenshot

Write down the values of x where y equals 0.

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6b
Sme Calculator1 mark

Show that

4 minus open parentheses x minus 2 close parentheses squared equals 4 x minus x squared

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6c
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Evaluate

space integral subscript 0 superscript 4 open parentheses 4 x minus x squared close parentheses space straight d x

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6d
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Write down the area of the region labelled R.

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7a
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Given  fraction numerator d squared y over denominator d x squared end fraction space equals 30 x and that when x equals 1 ,fraction numerator d y over denominator d x end fraction equals 17,  show that

fraction numerator d y over denominator d x end fraction equals 15 x squared plus 2

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7b
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Find an equation for y in terms of x, given that when x equals 2, y equals 40.

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8
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The diagram below shows part of the graph of  y equals x squared minus 4 x plus 3.

Find the area of the shaded region labelled R.

q11-8-1-integration-medium-a-level-maths-pure-screenshot

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9a
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Find the x-coordinates of the intercepts of the line with equation space y equals 2 space spaceand the curve with equation space y equals x squared minus 4 x plus 5.

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9b
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Evaluate

integral subscript 1 superscript 3 open parentheses x squared minus 4 x plus 5 close parentheses space straight d x

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9c
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The diagram below shows the graphs of space y equals 2 space spaceand  y equals x squared minus 4 x plus 5.
Find the exact area of the shaded region R.

q9-8-1-integration-medium-a-level-maths-pure-screenshot

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10a
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The diagram below shows the graphs of the line space y equals 6 minus x spaceand  the curve space y equals x squared.

GztGkQjs_q10-8-1-integration-easy-a-level-maths-pure-screenshot

Work out the x-coordinates of the points labelled P, Q and R.

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10b
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Work out the area of the shaded region.

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1a
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Use calculus to find

integral open parentheses x squared plus 5 minus 3 over x squared close parentheses space straight d x

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1b
Sme Calculator2 marks

Write down

integral 3 e to the power of 3 x end exponent space d x

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2
Sme Calculator5 marks

Use calculus to find the value of

integral subscript 4 superscript 9 fraction numerator x squared space plus space 1 over denominator square root of x end fraction space straight d x

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3
Sme Calculator4 marks

Find the equation of the curve passing through the point (-2, 3) and given by

y equals integral open parentheses 3 minus 2 x plus 4 over x squared close parentheses space straight d x

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4a
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Using the binomial expansion, or otherwise, show that

open parentheses 2 minus x close parentheses cubed equals 8 minus 12 x plus 6 x squared minus x cubed

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4b
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Hence, or otherwise, work out

integral open parentheses 2 minus x close parentheses cubed space straight d x

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5
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Given

integral subscript 1 superscript p left parenthesis 1 plus 1 over x squared right parenthesis space straight d x equals 15 over 4

find the value of the constant p, where p greater than 0.

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6
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The diagram below shows part of the graph of y equals 2 x plus 3 x squared minus 2 x cubed.

q5-8-1-integration-hard-a-level-maths-pure-screenshot

Find the area of the shaded region labelled R.

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7
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A function,space straight f open parentheses x close parentheses, has second derivative given by

space space straight f apostrophe apostrophe open parentheses x close parentheses equals 6 left parenthesis x minus 2 right parenthesis.

Given that straight f open parentheses 3 close parentheses equals 20, and straight f apostrophe open parentheses 2 close parentheses equals 8 comma find straight f open parentheses x close parentheses.

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8
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The diagram below shows part of the graph of y equals x left parenthesis x minus 1 right parenthesis left parenthesis x plus 2 right parenthesis.

Find the total area of the two shaded regions.

q7-8-1-integration-hard-a-level-maths-pure-screenshot

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9a
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The line with equation  5 y equals 14 minus 3 x  cuts the curve with equation  5 y equals 20 minus 2 x minus x squared at the points P and Q, as shown.

q9-8-1-integration-hard-a-level-maths-pure-screenshot

Find the x- and y-coordinates of the points P and Q.

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9b
Sme Calculator6 marks

Find the exact area of the region labelled R, giving your answer in the form  a over b, where a and b are integers to be found.

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10
Sme Calculator11 marks

The diagram below shows the graphs of space y equals x plus 2 space spaceand  y equals 10 x minus x squared minus 16.

q10-8-1-integration-hard-a-level-maths-pure-screenshot

Find the exact area of the shaded region.

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1
Sme Calculator3 marks

Use calculus to find

integral open parentheses 2 square root of x plus 5 x to the power of 1 third end exponent close parentheses space straight d x

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2
Sme Calculator5 marks

Use calculus to find the value of

integral subscript 2 superscript 4 fraction numerator x cubed plus cube root of x over denominator 2 square root of x end fraction space d x

giving your answer correct to 3 significant figures.

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3
Sme Calculator4 marks

Find the equation of the curve passing through the point (4, -8) and given by

y equals integral open parentheses fraction numerator 2 over denominator square root of x end fraction minus x minus 3 close parentheses space straight d x

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4a
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Show that


open parentheses 3 minus 1 half x close parentheses cubed equals 27 minus 27 over 2 x plus 9 over 4 x squared minus 1 over 8 x cubed

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4b
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Hence, or otherwise, work out

integral open parentheses 2 open parentheses 3 minus 1 half x close parentheses close parentheses cubed space space straight d x

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5
Sme Calculator5 marks

Given

integral subscript q superscript 4 q end superscript 5 x square root of x space straight d x equals 15 space 066

find the value of the constant q.

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6a
Sme Calculator4 marks

The diagram below shows part of the graph of y equals 4 x minus x cubed.

q6-8-1-integration-vh-a-level-maths-pure-screenshot

Find the area of the shaded region labelled R.

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6b
Sme Calculator1 mark

Without doing any additional calculation, explain why  integral subscript negative 2 end subscript superscript 2 space open parentheses 4 x minus x cubed space close parentheses straight d x  must be equal to zero.

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7
Sme Calculator6 marks

A function, straight f left parenthesis x right parenthesis, has second derivative given by

 space straight f apostrophe apostrophe open parentheses x close parentheses equals 2 open parentheses 18 x minus 5 close parentheses.

Given that left parenthesis 2 x minus 1 right parenthesis and left parenthesis 3 x plus 2 right parenthesis are factors of straight f left parenthesis x right parenthesis, find straight f left parenthesis x right parenthesis.

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8
Sme Calculator8 marks

The diagram below shows part of the graph of y equals x cubed minus 2 x squared minus x plus 2.

Find the total area of the two shaded regions.

q8-8-1-integration-vh-a-level-maths-pure-screenshot

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9
Sme Calculator11 marks

The diagram below shows the graphs of space 4 y equals 4 x plus 17 space space and  y equals 3 x squared minus 8 x minus 16.

Find the exact area of the shaded region.

q6-8-1-integration-vhard-a-level-maths-pure-screenshot

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10
Sme Calculator11 marks

The diagram below shows the graphs of  space y equals 16 minus 2 x space spaceand  y equals x cubed minus 15 x squared plus 66 x minus 80.

Find the total area of the shaded regions.

q7-8-1-integration-vhard-a-level-maths-pure-screenshot



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