Revision NotesExam QuestionsPast Papers
O LevelFurther MathsCambridge (CIE)Additional MathsExam QuestionsCalculusIntegration

Integration (Cambridge (CIE) O Level Additional Maths): Exam Questions

2 hours17 questions
Medium
Hard
Very Hard
1a
Sme Calculator4 marks

Giving your answer in its simplest form, find the exact value of 

integral subscript 0 superscript 4 fraction numerator 10 over denominator 5 x plus 2 end fraction d x

How did you do?

1b
Sme Calculator5 marks

integral subscript 0 superscript ln space 2 end superscript open parentheses straight e to the power of 4 x plus 2 end exponent close parentheses squared space straight d x

How did you do?

Did this page help you?

2
Sme Calculator5 marks

Find fraction numerator straight d over denominator straight d x end fraction open parentheses 16 minus x squared close parentheses to the power of 3 over 2 end exponentand hence evaluate the area enclosed by the curve y equals x square root of 16 minus x squared end root and the lines y space equals space 0 comma space x space equals 1 space and space x space equals space 3.

How did you do?

Did this page help you?

1
Sme Calculator6 marks

Find the exact value of integral subscript 2 superscript 4 open parentheses x plus 1 close parentheses squared over x squared straight d x.

How did you do?

Did this page help you?

2
Sme Calculator5 marks

Given

fraction numerator straight d over denominator straight d x end fraction open parentheses x cos space x close parentheses equals negative x sin space x plus cos space x

find the exact value of integral subscript 0 superscript straight pi over 6 end superscript space x space sin space x space straight d x.

How did you do?

Did this page help you?

3a
Sme Calculator7 marks

Given that integral subscript 1 superscript a open parentheses fraction numerator 2 over denominator 2 x space plus space 3 end fraction plus fraction numerator 3 over denominator 3 x space minus space 1 end fraction minus 1 over x close parentheses straight d x space equals space ln space 2.4 where a space greater than space 1 , find the value of a.

How did you do?

3b
Sme Calculator4 marks

(i) Find fraction numerator straight d over denominator straight d x end fraction open parentheses 6 space sin cubed space k x close parentheses, where k is a constant.

(ii) Hence find integral left parenthesis sin squared space 2 x space cos space 2 x right parenthesis d x.

How did you do?

Did this page help you?

4
Sme Calculator5 marks

Find  integral subscript 3 superscript 5 open parentheses fraction numerator 1 over denominator x minus 1 end fraction space minus space 1 over open parentheses x minus 1 close parentheses squared close parentheses space straight d x, giving your answer in the form a plus ln space b, where a and b are rational numbers.

How did you do?

Did this page help you?

5
Sme Calculator7 marks

A curve is such that fraction numerator straight d squared y over denominator straight d x squared end fraction equals sin open parentheses 6 x minus pi over 2 close parentheses. Given that fraction numerator straight d y over denominator straight d x end fraction equals 1 half at the point open parentheses pi over 4 comma fraction numerator 13 pi over denominator 12 end fraction close parentheses on the curve, find the equation of the curve.

How did you do?

Did this page help you?

6a
Sme Calculator2 marks
q10-0606-s20-qp-21-additional-maths

The diagram shows part of the graphs of y equals 4 x to the power of 2 over 3 end exponent and y space equals space left parenthesis x minus 3 right parenthesis squared .The graph of y space equals space left parenthesis x minus 3 right parenthesis squared meets the x-axis at the point A left parenthesis a comma space 0 right parenthesis and the two graphs intersect at the point B left parenthesis b comma space 4 right parenthesis.

Find the value of a and of b.

How did you do?

6b
Sme Calculator5 marks

Find the area of the shaded region.

How did you do?

Did this page help you?

7a
Sme Calculator3 marks
q6-0606-s20-qp-12-additional-maths

The diagram shows the straight line 2 x plus y space equals negative 5 and part of the curve x y plus 3 space equals space 0. The straight line intersects the x-axis at the point A and intersects the curve at the point B. The point C lies on the curve. The point D has coordinates (1, 0). The line C D is parallel to the y-axis.

Find the coordinates of each of the points A and B.

How did you do?

7b
Sme Calculator6 marks

Find the area of the shaded region, giving your answer in the form p plus ln space q, where p and q are positive integers.

How did you do?

Did this page help you?

8
Sme Calculator8 marks
q9-0606-s20-qp-13-additional-maths

The diagram shows part of the curve x y space equals space 2 intersecting the straight line y space equals space 5 x minus 3 at the point A.
The straight line meets the x-axis at the point B. The point C lies on the x-axis and the point D lies on the curve such that the line C D has equation x space equals space 3. Find the exact area of the shaded region, giving your answer in the form p plus ln space q, where p and q are constants.

How did you do?

Did this page help you?

9a
Sme Calculator6 marks

(i) Given that straight f open parentheses x close parentheses equals fraction numerator 1 over denominator cos space x end fraction, show that straight f apostrophe left parenthesis x right parenthesis space equals space tan space x space sec space x.

[3]

(ii) Hence find integral open parentheses 3 space tan space x space sec space x minus fourth root of e to the power of 3 x end exponent end root close parentheses space straight d x.

[3]

How did you do?

9b
Sme Calculator5 marks

Given that integral subscript 2 superscript 5 fraction numerator p over denominator p x plus 10 end fraction straight d x equals ln 2, find the value of the positive constant p.

How did you do?

Did this page help you?

10a
Sme Calculator6 marks

Given that integral subscript 1 superscript a open parentheses 1 over x minus fraction numerator 1 over denominator 2 x plus 3 end fraction close parentheses space straight d x equals ln 3, where a greater than 0, find the exact value of a, giving your answer in simplest surd form.

How did you do?

10b
Sme Calculator5 marks

Find the exact value of integral subscript 0 superscript straight pi over 3 end superscript open parentheses sin open parentheses 2 x plus straight pi over 3 close parentheses minus 1 plus cos space 2 x close parentheses space straight d x.

How did you do?

Did this page help you?

1a
Sme Calculator1 mark

Show that fraction numerator 1 over denominator x plus 1 end fraction plus fraction numerator 2 over denominator 3 x plus 10 end fraction can be written as fraction numerator 5 x plus 12 over denominator 3 x squared plus 13 x plus 10 end fraction

How did you do?

1b
Sme Calculator9 marks
q10b-0606-w20-qp-13-additional-maths

The diagram shows part of the curve y equals fraction numerator 5 x plus 12 over denominator 3 x squared plus 13 x plus 10 end fraction, the line x equals 2 and a straight line of gradient 1. The curve intersects the y-axis at the point P. The line of gradient 1 passes through P and intersects the x-axis at the point Q. Find the area of the shaded region, giving your answer in the form a plus 2 over 3 space ln open parentheses b square root of 3 close parentheses, where a and b are constants.

How did you do?

Did this page help you?

2
Sme Calculator9 marks
q12-2025-specimen-paper-2-cie-igcse-additional-maths

The diagram shows part of the curve y space equals space left parenthesis 9 space – space x right parenthesis left parenthesis x space – space 3 right parenthesis and the line y space equals space k space – space 3, where space k space greater than space 3.
The line through the maximum point of the curve, parallel to the y-axis, meets the x-axis at A.
The curve meets the x-axis at B, and the line y space equals space k space – space 3 meets the curve at the point C left parenthesis k comma space k space – space 3 right parenthesis .

Find the area of the shaded region.

How did you do?

Did this page help you?

3a
Sme Calculator2 marks

Show that fraction numerator 3 over denominator 2 x minus 3 end fraction plus fraction numerator 3 over denominator 2 x plus 3 end fraction can be written as fraction numerator 12 x over denominator 4 x squared minus 9 end fraction.

How did you do?

3b
Sme Calculator3 marks

Hence find integral fraction numerator 12 x over denominator 4 x squared minus 9 end fraction straight d x, giving your answer as a single logarithm and an arbitrary constant.

How did you do?

3c
Sme Calculator4 marks

Given that integral subscript 2 superscript a fraction numerator 12 x over denominator 4 x squared minus 9 end fraction straight d x equals ln space 5 square root of 5, where a space greater than space 2, find the exact value of a.

How did you do?

Did this page help you?

4
Sme Calculator8 marks

A curve is such that fraction numerator straight d squared y over denominator straight d x squared end fraction equals 5 cos space 2 x. This curve has a gradient of 3 over 4 at the point open parentheses negative straight pi over 12 comma fraction numerator 5 straight pi over denominator 4 end fraction close parentheses. Find the equation of this curve.

How did you do?

Did this page help you?

5a
Sme Calculator5 marks

The gradient of the normal to a curve at the point left parenthesis x comma space y right parenthesis is given by fraction numerator x over denominator x plus 1 end fraction

Given that the curve passes through the point left parenthesis 1 comma space 4 right parenthesis, show that its equation is y space equals space 5 minus ln space x minus x.

How did you do?

5b
Sme Calculator3 marks

Find, in the form y space equals space m x plus c, the equation of the tangent to the curve at the point where x space equals space 3.

How did you do?

Did this page help you?