ASMathsCambridge (CIE)Pure 1Exam QuestionsIntegrationIntegration

Integration (Cambridge (CIE) AS Maths: Pure 1): Exam Questions

Exam code: 9709

2 hours28 questions
Easy
Medium
Hard
Very Hard
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.

How did you do?

2
Sme Calculator4 marks

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
Sme Calculator3 marks

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.

How did you do?

4a
Sme Calculator2 marks

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.

How did you do?

4b
Sme Calculator3 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.

How did you do?

5a
Sme Calculator3 marks

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

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5b
Sme Calculator3 marks

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.

How did you do?

6
Sme Calculator4 marks

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.

How did you do?

7
Sme Calculator4 marks

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.

How did you do?

8a
Sme Calculator3 marks

By first writing 1 over x squared as x to the power of negative 2 end exponent, use calculus to show that

      integral subscript 1 superscript a 1 over x squared d x equals 1 minus 1 over a

where a space greater than space 1 is a constant.

 

How did you do?

8b
Sme Calculator2 marks

Find the value of the integral in part (a) when

(i) a space equals space 10 comma

(ii) a space equals space 1000 comma

(iii) a space equals space 1 space 000 space 000.

How did you do?

8c
Sme Calculator1 mark

By considering your answers to part (b), suggest what number the value of the integral in part (a) will approach as a space rightwards arrow infinity.

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1
Sme Calculator2 marks

Use calculus to find

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

How did you do?

2
Sme Calculator2 marks

Evaluate

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

How did you do?

3
Sme Calculator3 marks

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

How did you do?

4a
Sme Calculator2 marks

Show that

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

How did you do?

4b
Sme Calculator2 marks

Hence, or otherwise, work out

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

How did you do?

5
Sme Calculator4 marks

Given

      integral subscript k superscript 5 open parentheses 2 x minus 1 close parentheses d x equals 20

find the value of the positive constant k.

How did you do?

6a
Sme Calculator4 marks

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

How did you do?

6b
Sme Calculator3 marks

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

How did you do?

7a
Sme Calculator3 marks

Use calculus to show that

      integral subscript 1 superscript a 2 over x cubed d x equals 1 minus 1 over a squared

where a space greater than space 1 is a constant.

How did you do?

7b
Sme Calculator2 marks

By first considering the limit of 1 over a squared as a space rightwards arrow space infinity,  determine the value of

      integral subscript 1 superscript infinity 2 over x cubed d x

How did you do?

1
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

How did you do?

2
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

How did you do?

3a
Sme Calculator3 marks

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

How did you do?

3b
Sme Calculator3 marks

Hence, or otherwise, work out

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

How did you do?

4
Sme Calculator5 marks

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.

How did you do?

5
Sme Calculator5 marks

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.

How did you do?

6a
Sme Calculator4 marks

Use calculus to find the exact value of each of the following improper integrals:

      integral subscript 7 superscript infinity 5 over x squared d x

How did you do?

6b
Sme Calculator4 marks

      integral subscript 0 superscript 18 fraction numerator 2 over denominator square root of x end fraction d x

How did you do?

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

How did you do?

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.

How did you do?

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

How did you do?

4a
Sme Calculator3 marks

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

How did you do?

4b
Sme Calculator3 marks

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

How did you do?

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.

How did you do?

6
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.

How did you do?

7a
Sme Calculator5 marks

Given that

      integral subscript p superscript infinity fraction numerator 3 over denominator x square root of x end fraction d x equals square root of 3 

where p is a real constant, find the value of p.

 

How did you do?

7b
Sme Calculator5 marks

Given that

      integral subscript 0 superscript 50 space fraction numerator q plus 3 x over denominator square root of x end fraction d x equals 750 square root of 2

where q is a real constant, find the value of q.

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