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Integration (Edexcel A Level Maths: Pure): Exam Questions

5 hours53 questions
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1
Sme Calculator3 marks

Find

(i) integral 2 x space straight d x

(ii) integral 6 x squared space straight d x

(iii) integral 1 half x to the power of 1 half end exponent space straight d x

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2
Sme Calculator4 marks

Find the value of

(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
Sme Calculator2 marks

Find

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

writing each term in simplest form.

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4
Sme Calculator2 marks

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

writing your answer in simplest form.

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5
Sme Calculator2 marks

Find the value of

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

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6a
Sme Calculator2 marks

Write

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

in the form  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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6b
Sme Calculator2 marks

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

writing your answer in simplest form.

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7a
Sme Calculator2 marks

Find

integral open parentheses 5 x to the power of 4 plus 6 x squared plus 2 x plus 3 close parentheses space straight d x

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7b
Sme Calculator2 marks

Given that

  • space straight f apostrophe left parenthesis x right parenthesis equals 5 x to the power of 4 plus 6 x squared plus 2 x plus 3

  • the curve space y equals straight f left parenthesis x right parenthesis passes through the point open parentheses 1 comma space 10 close parentheses

find straight f left parenthesis x right parenthesis, giving your answer in simplest form.

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8
Sme Calculator4 marks

A curve, C, is defined by the following:

  • fraction numerator straight d y over denominator straight d x end fraction equals 2 x cubed minus x

  • C passes through the point open parentheses 2 comma space minus 2 close parentheses

Show that the equation of C can be written in the form

space 2 y equals x to the power of 4 minus x squared plus k

where k is a constant to be found.

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9a
Sme Calculator1 mark

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 shaded below.

Graph of y = 9 - x² with a shaded area between x = 1 and x = 2 and the x axis, bounded by dashed lines.

Write down a definite integral that represents this area.

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9b
Sme Calculator2 marks

Use algebraic integration to find the exact area of the shaded region in part (a).

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10
Sme Calculator4 marks

The diagram below shows the curve with equation y equals 7 x minus x squared minus 6 passing through the points open parentheses 1 comma space 0 close parentheses and open parentheses 6 comma space 0 close parentheses on the x-axis.

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

Use algebraic integration to find the exact area of the shaded region bounded by the curve and the x-axis.

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11
Sme Calculator4 marks

The diagram below shows the curve with equation y equals x squared minus 8 x plus 12 passing through the points open parentheses 2 comma space 0 close parentheses and open parentheses 6 comma space 0 close parentheses on the x-axis.

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

Use algebraic integration to find the exact area of the shaded region.

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12
Sme Calculator4 marks

A curve C has the equation y equals straight f open parentheses x close parentheses

Given that

  • straight f apostrophe open parentheses x close parentheses equals 3 minus 2 x plus 6 x squared

  • the curve passes through the point open parentheses 4 comma space 64 close parentheses

find the equation of the curve, giving your answer in simplest form.

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13a
Sme Calculator1 mark

Simplify 

space x minus open parentheses x squared minus 8 x plus 18 close parentheses

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13b
Sme Calculator4 marks

The diagram below shows the curve with equation y equals x squared minus 8 x plus 18 intersecting the straight line y equals x at the points open parentheses 3 comma space 3 close parentheses and open parentheses 6 comma space 6 close parentheses.

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

Use algebraic integration to find the exact area of the shaded region.

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14
Sme Calculator4 marks

The finite region R, shown in the figure below, is bounded by the curve with equation y equals x squared minus 4 x plus 3 and the x-axis.

Use algebraic integration to find the exact area of R.

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

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15
Sme Calculator4 marks

The figure below shows the finite shaded region, R, bounded by the curve y equals 2 x plus 3 x squared minus 2 x cubed and the x-axis.

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

Two x-intercepts of y equals 2 x plus 3 x squared minus 2 x cubed are shown, 0 and 2.

Use algebraic integration to find the exact area of R.

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16
Sme Calculator3 marks

Find

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

writing each term in simplest form.

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14 marks

Find

integral open parentheses 8 x cubed minus fraction numerator 3 over denominator 2 square root of x end fraction plus 5 close parentheses space straight d x

giving your answer in simplest form.

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24 marks

Find

integral fraction numerator x to the power of 1 half end exponent open parentheses 2 x minus 5 close parentheses over denominator 3 end fraction straight d x

writing each term in simplest form.

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34 marks

Find

integral fraction numerator 3 x to the power of 4 minus 4 over denominator 2 x cubed end fraction straight d x

writing your answer in simplest form.

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4a
Sme Calculator2 marks

The curve C has equation y equals straight f open parentheses x close parentheses

The curve

  • passes through the point P open parentheses 3 comma space minus 10 close parentheses

  • has a turning point at P

Given that

fraction numerator straight d y over denominator straight d x end fraction equals 2 x cubed minus 9 x squared plus 5 x plus k

where k is a constant,

show that k equals 12.

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4b
Sme Calculator3 marks

Hence find the coordinates of the point where C crosses the y-axis.

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5
Sme Calculator4 marks

Find the value of the constant k, 0 less than k less than 9, such that

integral subscript k superscript 9 fraction numerator 6 over denominator square root of x end fraction straight d x equals 20

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6
Sme Calculator4 marks

Find the value of the positive constant k such that

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

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7a
Sme Calculator3 marks

Find

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

giving your answer in simplest form.

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7b
Sme Calculator2 marks

Write down

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

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8
Sme Calculator6 marks

The figure below shows a sketch of the line y equals 2 and the curve with equation y equals x squared minus 4 x plus 5.

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

Use algebraic integration to find the exact area of R.

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9
Sme Calculator6 marks

The curve with equation y equals x squared and the straight line with equation y equals 6 minus x are shown in the figure below.

The finite region bounded by the curve, the line and the x-axis is shaded.

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

Use algebraic integration to find the exact area of the shaded region.

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10
Sme Calculator5 marks

Find the exact 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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11
Sme Calculator5 marks

The finite region R, shown in the figure below, is bounded by the curve with equationspace y equals 4 minus open parentheses x minus 2 close parentheses squared and the x-axis.

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

The x-intercepts of the curve space y equals 4 minus open parentheses x minus 2 close parentheses squared are 0 and 4.

Use algebraic integration to find the exact area of R.

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12
Sme Calculator3 marks

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

giving your answer in simplest form.

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13
Sme Calculator4 marks

A curve has the equation y equals straight f open parentheses x close parentheses

Given that

  • straight f apostrophe open parentheses x close parentheses equals 3 minus 2 x plus 4 over x squared

  • the curve passes through the point open parentheses negative 2 comma space 3 close parentheses

find the equation of the curve, giving your answer in simplest form.

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14
Sme Calculator3 marks

Find the integer value of b such that

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

where k is a constant.

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15a
Sme Calculator3 marks

A curve has the equation y equals straight f open parentheses x close parentheses

Given that

  • fraction numerator straight d squared y over denominator straight d x squared end fraction space equals 30 x

  • fraction numerator d y over denominator d x end fraction equals 17 when x equals 1

find an expression for fraction numerator d y over denominator d x end fraction

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15b
Sme Calculator3 marks

Given that

  • y equals 40 when x equals 2

find the equation of the curve, y equals straight f open parentheses x close parentheses, giving your answer in simplest form.

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16
Sme Calculator6 marks

The figure below shows a sketch of the curve with equation y equals x left parenthesis x minus 1 right parenthesis left parenthesis x plus 2 right parenthesis.

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

Use algebraic integration to find the total area of the shaded regions.

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17
Sme Calculator4 marks

The curve C has the equation y equals straight f open parentheses x close parentheses

Given that

  • straight f apostrophe open parentheses x close parentheses equals fraction numerator 2 over denominator square root of x end fraction minus x minus 3

  • C passes through the point open parentheses 4 comma negative 8 close parentheses

find the equation of C, giving your answer in simplest form.

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18a
Sme Calculator4 marks

The curve with equation y equals 4 x minus x cubed is shown in the figure below.

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

The region bounded by the curve and the x-axis, where x less or equal than 0 and y less or equal than 0, is shaded.

Using algebraic integration, find the exact area of R.

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18b
Sme Calculator1 mark

Write down the value of 

integral subscript negative 2 end subscript superscript 2 space open parentheses 4 x minus x cubed space close parentheses straight d x

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15 marks

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Graph showing the shaded region R above the curve y=4(x^2)+3, below the horizontal line y=23, and to the right of the y-axis, with labelled x and y axes and origin point O.
Figure 2

The finite region R, shown shaded in Figure 2, is bounded by the curve with equation y equals 4 x squared plus 3, the y-axis and the line with equation y equals 23

Show that the exact area of R is k square root of 5 where k is a rational constant to be found.

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26 marks

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Graph of a curve on xy-axes for positive values of x, with a shaded region labelled R below the x-axis enclosed by the curve and the x-axis.
Figure 3

Figure 3 shows a sketch of part of a curve with equation

y equals fraction numerator open parentheses x minus 2 close parentheses open parentheses x minus 4 close parentheses over denominator 4 square root of x end fraction space space space space space space space space space space space space space x greater than 0

The region R, shown shaded in Figure 3, is bounded by the curve and the x-axis.

Find the exact area of R, writing your answer in the form a square root of 2 plus b , where a and b are constants to be found.

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3a5 marks
Graph showing a curve C and a tangent line l, that touches the curve at point P. The region bounded by C, l and the coordinate axes is shaded and labelled R.
Figure 2

In this question your must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Figure 2 shows a sketch of part of the curve C with equation

table row cell y equals 1 third x squared minus 2 square root of x plus 3 end cell blank cell x greater or equal than 0 end cell end table

The point P lies on C and has x coordinate 4

The line l is tangent to C at P.

Show that l has equation

13 x minus 6 y minus 26 equals 0

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3b
Sme Calculator5 marks

The region R, shown shaded in Figure 2 is bounded by the y-axis, the curve C, the line l and the x-axis.

Find the exact area of R.

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4
Sme Calculator5 marks

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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5
Sme Calculator5 marks

Find

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

writing each term in simplest form.

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6
Sme Calculator5 marks

Find the value of the constant p, where p greater than 0, such that

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

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7
Sme Calculator5 marks

The function straight f open parentheses x close parentheses has the following properties

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

  • straight f apostrophe open parentheses 2 close parentheses equals 8

  • straight f open parentheses 3 close parentheses equals 20

Find straight f open parentheses x close parentheses, giving your answer in simplest form.

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8
Sme Calculator6 marks

The figure below shows the straight line with equation 5 y equals 14 minus 3 x and the curve with equation 5 y equals 20 minus 2 x minus x squared.

The shaded region R is bounded by the curve and the straight line.

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

Use algebraic integration to find the exact area of R.

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9
Sme Calculator7 marks

The figure below shows the straight line y equals x plus 2 and the curve with equation y equals 10 x minus x squared minus 16.

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

Use algebraic integration to find the exact area of the shaded region.

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10
Sme Calculator8 marks

The curve with equation y equals x cubed minus 2 x squared minus x plus 2 is shown in the figure below.

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

Use algebraic integration to find the total area of the shaded regions.

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11a3 marks

A curve C has equation y equals straight f open parentheses x close parentheses where

straight f left parenthesis x right parenthesis equals – 3 x squared plus 12 x plus 8

Write straight f open parentheses x close parentheses in the form

a open parentheses x plus b close parentheses squared plus c

where a, b and c are constants to be found.

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11b2 marks

The curve C has a maximum turning point at M.

Find the coordinates of M.

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11c
Sme Calculator5 marks
Graph with an 'n-shaped' parabola labelled C, with a vertex maximum point in the first quadrant labelled M. A horizontal line l is tangent to the parabola at point M. The region in the first quadrant enclosed by the parabola, line l, and the positive y-axis, is shaded and labelled R. The x- and y-axes are shown and labelled.
Figure 3

Figure 3 shows a sketch of the curve C.

The line l passes through M and is parallel to the x-axis.

The region R, shown shaded in Figure 3, is bounded by C, l and the y-axis.

Using algebraic integration, find the area of R.

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1
Sme Calculator6 marks

A curve has equation y equals straight f open parentheses x close parentheses, space x greater or equal than 0

Given that

  • straight f to the power of apostrophe open parentheses x close parentheses equals 4 x plus a square root of x plus b, where a and b are constants

  • the curve has a stationary point at open parentheses 4 comma space 3 close parentheses

  • the curve meets the y-axis at negative 5

find straight f open parentheses x close parentheses, giving your answer in simplest form.

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2a4 marks
Graph with a positive cubic curve. The intersects the x-axis at 3 points, once on the negative x-axis, at (0, 0) and once on the positive x-axis. The region below the curve and above the x-axis between the first two points of intersection is shaded and labelled R1. The region above the curve and below the x-axis between the second point of intersection and the point b on the positive x-axis is shaded and labelled R2.
Figure 2

Figure 2 shows a sketch of part of the curve with equation y equals x open parentheses x plus 2 close parentheses open parentheses x minus 4 close parentheses.

The region R subscript 1 shown shaded in Figure 2 is bounded by the curve and the negative x-axis.

Show that the exact area of R subscript 1 is 20 over 3

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2b4 marks

The region R subscript 2 also shown shaded in Figure 2 is bounded by the curve, the positive x-axis and the line with equation x equals b, where b is a positive constant and 0 less than b less than 4

Given that the area of R subscript 1 is equal to the area of R subscript 2 verify that b satisfies the equation

open parentheses b plus 2 close parentheses squared open parentheses 3 b squared minus 20 b plus 20 close parentheses equals 0

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2c2 marks

The roots of the equation 3 b squared minus 20 b plus 20 equals 0 are 1.225 and 5.442 to 3 decimal places.

The value of b is therefore 1.225 to 3 decimal places.

Explain, with the aid of a diagram, the significance of the root 5.442

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3a4 marks

In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Graph with curve C and line l intersecting at point P(5, -13). Shaded region R is beneath the curve and above the line, with axes labelled x and y.

Figure 2 shows a sketch of part of the curve C with equation

y equals x cubed minus 10 x squared plus 27 x minus 23

The point P open parentheses 5 comma space minus 13 close parentheses lies on C

The line l is tangent to C at P

Use differentiation to find the equation of l, giving your answer in the form y equals m x plus c where m and c are integers to be found.

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3b
Sme Calculator1 mark

Hence verify that l meets C again on the y-axis.

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3c4 marks

The finite region R, shown shaded in Figure 2, is bounded by the curve C and the line l.

Use algebraic integration to find the exact area of R.

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4
Sme Calculator6 marks

Given that

integral open parentheses 2 open parentheses 3 minus 1 half x close parentheses close parentheses to the power of p space straight d x equals negative 1 fourth x to the power of 4 plus...

where p is a positive integer and negative 1 fourth x to the power of 4 is the term with the highest power of x, find fully

integral open parentheses 2 open parentheses 3 minus 1 half x close parentheses close parentheses to the power of p space straight d x

writing each term in simplest form.

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5
Sme Calculator5 marks

Find the value of the constant q such that

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

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6
Sme Calculator6 marks

A function, straight f left parenthesis x right parenthesis

  • has a factor of open parentheses 2 x minus 1 close parentheses

  • has a factor of open parentheses 3 x plus 2 close parentheses

  • has a second derivative of straight f apostrophe apostrophe open parentheses x close parentheses equals 2 open parentheses 18 x minus 5 close parentheses

Find straight f left parenthesis x right parenthesis

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7
Sme Calculator8 marks

The straight line with equation 4 y equals 4 x plus 17 and the curve with equation y equals 3 x squared minus 8 x minus 16 are shown in the figure below.

The region bounded by the straight line, the curve and the x-axis is shaded.

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

Use algebraic integration to find the exact area of the shaded region.

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8
Sme Calculator9 marks

The figure below shows a sketch of the line y equals 16 minus 2 x and the curve y equals x cubed minus 15 x squared plus 66 x minus 80.

Use algebraic integration to find the exact total area of the shaded regions.

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

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