A LevelMathsAQAPureExam QuestionsIntegrationFurther Integration

Further Integration (AQA A Level Maths: Pure): Exam Questions

Exam code: 7357

3 hours34 questions
Easy
Medium
Hard
Very Hard
1
Sme Calculator2 marks

A student is estimating the area bounded by the curve space y equals straight f left parenthesis x right parenthesis, the x-axis and the lines x equals a and x equals b.

The student intends to estimate the area by using trapezia of equal width.

q1-8-2-further-integration-easy-a-level-maths-pure-screenshots

Add to the diagram above to show how the student can use 4 trapezia to estimate the area.

How did you do?

23 marks

Find

(i)

integral 1 over x space straight d x

(ii)

integral subscript 0 superscript 1 straight e to the power of x space straight d x

(iii)

integral 3 cos theta space straight d theta

How did you do?

3a2 marks

Use a suitable substitution to show that

integral 3 cos open parentheses 3 x plus 2 close parentheses space straight d x equals integral cos u space straight d u

How did you do?

3b1 mark

Hence find

integral 3 cos space open parentheses 3 x plus 2 close parentheses space straight d x

How did you do?

4a1 mark

Find

integral sin x space straight d x

How did you do?

4b
Sme Calculator2 marks

Show that

integral subscript 1 superscript 4 1 over x space straight d x equals ln space 4

How did you do?

4c1 mark

Find

integral 7 straight e to the power of 7 x end exponent space straight d x

How did you do?

5
Sme Calculator6 marks

Find

(i)

integral 8 open parentheses 2 x minus 1 close parentheses cubed space straight d x

(ii)

integral subscript 0 superscript pi over 4 end superscript sin space 2 x space straight d x

(iii)

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

How did you do?

6a1 mark

Given the identity cos space 2 A identical to 1 minus 2 space sin squared A, show that

sin squared A identical to 1 half left parenthesis 1 minus cos space 2 A right parenthesis

How did you do?

6b
Sme Calculator3 marks

Hence find the exact value of

integral subscript pi over 2 end subscript superscript pi sin squared x space straight d x

How did you do?

73 marks

Show that

integral subscript 1 superscript 2 straight e to the power of 3 x plus 2 end exponent space d x equals 1 third straight e to the power of 5 open parentheses straight e cubed minus 1 close parentheses

How did you do?

8a2 marks

Find

integral cos space 2 x space straight d x

How did you do?

8b
Sme Calculator3 marks

Show that

integral subscript 0 superscript 2 open parentheses 3 x minus 1 close parentheses cubed space straight d x equals 52

How did you do?

8c1 mark

Find

integral straight e to the power of 5 x end exponent space straight d x

How did you do?

9a2 marks

Find

integral 5 left parenthesis straight e to the power of 5 x end exponent minus straight e to the power of negative 5 x end exponent right parenthesis space straight d x

How did you do?

9b2 marks

Find

integral open parentheses sin space x plus cos space x close parentheses space straight d x

How did you do?

9c
Sme Calculator3 marks

Show that

integral subscript negative 8 end subscript superscript negative 2 end superscript 1 over x space straight d x equals ln open parentheses 1 fourth close parentheses

How did you do?

12 marks

Find

integral fraction numerator 4 x over denominator 2 x squared plus 5 end fraction space straight d x

How did you do?

24 marks

Show that

integral subscript 2 superscript 6 4 open parentheses 4 x plus 1 close parentheses to the power of 1 half end exponent d x equals 196 over 3

How did you do?

33 marks

Find

integral negative 15 sin open parentheses 5 x minus 2 close parentheses space straight d x

How did you do?

43 marks

Find

integral negative 3 x sin x space straight d x

How did you do?

56 marks

Use the substitution u equals x plus 4 to show that

integral subscript 1 superscript 2 fraction numerator x over denominator x plus 4 end fraction space straight d x equals 1 plus 4 ln open parentheses 5 over 6 close parentheses

How did you do?

63 marks

Find

integral fraction numerator 3 x squared plus 2 over denominator 2 x cubed plus 4 x end fraction space straight d x

How did you do?

7a2 marks

Find

integral 2 sin x cos x space straight d x

How did you do?

7b
Sme Calculator4 marks

Use algebraic integration to find the exact value of

integral subscript 1 superscript 3 open parentheses 4 x plus 1 close parentheses to the power of 5 space straight d x

How did you do?

8a2 marks

Show that

fraction numerator 11 over denominator left parenthesis 2 x minus 3 right parenthesis left parenthesis x plus 4 right parenthesis end fraction

can be written in the form

fraction numerator A over denominator 2 x minus 3 end fraction plus fraction numerator B over denominator x plus 4 end fraction

where A and B are constants to be found.

How did you do?

8b4 marks

Hence find

integral fraction numerator 11 over denominator left parenthesis 2 x minus 3 right parenthesis left parenthesis x plus 4 right parenthesis end fraction space straight d x

writing your answer in the form

ln vertical line straight f open parentheses x close parentheses vertical line plus c

where straight f open parentheses x close parentheses is a function you should find and c is a constant.

How did you do?

9
Sme Calculator6 marks

The figure below shows a sketch of the curves with equations y equals x squared minus 3 x plus 4 and y equals 4 minus x squared plus 2 x.

The shaded region R is bounded by the two curves.

q11-8-2-further-integration-medium-a-level-maths-pure-screenshot

Use algebraic integration to find the exact area of R.

How did you do?

14 marks

Use the substitution u equals 2 plus ln space x to show that

integral fraction numerator 1 over denominator x open parentheses 2 plus ln space x close parentheses cubed end fraction space straight d x equals fraction numerator negative 1 over denominator 2 open parentheses 2 plus ln space x close parentheses squared end fraction plus c

where c is a constant.

How did you do?

23 marks

Find 

integral square root of 1 plus cot squared x end root space d x

How did you do?

36 marks

Use algebraic integration to show that

integral subscript pi over 4 end subscript superscript pi over 2 end superscript cos squared theta space d theta equals pi over 8 minus 1 fourth

How did you do?

4
Sme Calculator3 marks

The figure below shows a sketch of the curve with equation y equals 1 plus 2 x minus 1 fourth x squared where

  • the point  P left parenthesis x comma space y right parenthesis lies on the curve

  • the shaded rectangle shown has width delta x and height y

q1-8-2-further-integration-hard-a-level-maths-pure-screenshot

By expressing the series limit as a suitable integral, show that

limit as delta x rightwards arrow 0 of space sum from x equals 1 to 8 of open parentheses space 1 plus 2 x minus 1 fourth x squared close parentheses space delta x equals 329 over 12

How did you do?

57 marks

Use algebraic integration to show that

integral subscript pi over 6 end subscript superscript pi over 3 end superscript space fraction numerator 2 minus 2 cos squared theta space over denominator sin space 2 theta end fraction space straight d theta equals 1 half ln space 3 space

How did you do?

65 marks

Use algebraic integration to show that

integral subscript 0 superscript 1 fraction numerator 3 x straight e to the power of negative 3 x squared end exponent over denominator 3 minus 2 straight e to the power of negative 3 x squared end exponent end fraction space straight d x equals 1 fourth ln left parenthesis 3 minus 2 straight e to the power of negative 3 end exponent right parenthesis

How did you do?

76 marks

Use algebraic integration to show that

integral subscript pi over 2 end subscript superscript fraction numerator 5 pi over denominator 6 end fraction end superscript space fraction numerator 2 space cos x over denominator 1 minus cos 2 x end fraction space straight d x equals negative 1

How did you do?

8a5 marks

Use algebraic integration to show that

integral open parentheses 2 x squared minus 1 close parentheses straight e to the power of x space straight d x equals open parentheses p x squared plus q x plus r close parentheses straight e to the power of x plus c

where p, q and r are integers to be found and c is a constant.

How did you do?

8b4 marks

Show that

integral ln x space straight d x equals x ln x minus x plus c

where c is a constant.

How did you do?

9a3 marks

Use algebraic integration to find

integral 5 cos squared 4 x sin 4 x space straight d x

How did you do?

9b3 marks

Use algebraic integration to find

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

How did you do?

10a3 marks

Use algebraic integration to find

integral 6 x squared straight e to the power of x cubed end exponent space straight d x

How did you do?

10b3 marks

Use algebraic integration to find

integral open parentheses 16 minus 32 x close parentheses sin open square brackets open parentheses 4 x minus 2 close parentheses squared close square brackets space straight d x

How did you do?

11a6 marks

Use algebraic integration to find

integral x squared sin space 3 x space straight d x

How did you do?

11b
Sme Calculator4 marks

Use algebraic integration to find

integral fraction numerator ln space x over denominator space x cubed end fraction space straight d x

How did you do?

16 marks

Use the substitution x equals cos space theta to show that

integral subscript fraction numerator 1 over denominator square root of 2 end fraction end subscript superscript fraction numerator square root of 3 over denominator 2 end fraction end superscript fraction numerator 1 over denominator square root of 1 minus x squared end root space end fraction straight d x equals pi over 12

How did you do?

2a4 marks

Prove that

integral tan space k x space straight d x equals 1 over k ln space open vertical bar sec space k x close vertical bar plus c

where k and c are constants.

How did you do?

2b5 marks

Use algebraic integration to show that

integral subscript pi over 18 end subscript superscript pi over 9 end superscript fraction numerator cosec squared 3 theta space over denominator 3 cot 3 theta end fraction straight d theta equals a ln 3

where a is a rational number to be found.

How did you do?

38 marks

Show that

integral fraction numerator 8 x squared minus 8 x minus 1 over denominator left parenthesis 4 x squared minus 1 right parenthesis left parenthesis x minus 2 right parenthesis end fraction space straight d x equals ln open parentheses A vertical line x minus 2 vertical line square root of vertical line 4 x squared minus 1 vertical line end root close parentheses

where A is a constant.

How did you do?

46 marks

Show that

integral straight e to the power of x sin x space straight d x equals 1 half straight e to the power of x open parentheses sin x minus cos x close parentheses plus c

where c is a constant.

How did you do?

5a1 mark

The figure below shows a sketch of the curves with equations y equals square root of 25 minus x squared end root     and y equals 6 x minus x squared minus 5.

The finite regions bounded by the two curves are shaded.

q9-8-2-further-integration-veryhard-a-level-maths-pure-screenshot

Show that the x-coordinates of the points of intersection are x equals 3, x equals 4 and x equals 5.

[You do not need to solve an equation in x.]

How did you do?

5b8 marks

Use the substitution x equals 5 sin u to show that

integral square root of 25 minus x squared end root space straight d x equals fraction numerator 25 arcsin open parentheses x over 5 close parentheses plus x square root of 25 minus x squared end root over denominator 2 end fraction plus c

where c is a constant.

How did you do?

5c5 marks

Hence show that the exact area of the shaded regions is

fraction numerator 25 pi over denominator 4 end fraction minus 4 minus 25 over 2 open parentheses 2 arcsin open parentheses 4 over 5 close parentheses minus arcsin open parentheses 3 over 5 close parentheses close parentheses

How did you do?