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O LevelAdditional MathsCambridgeExam QuestionsCalculusIntegration

Integration (Cambridge O Level Additional Maths)

Exam Questions

2 hours17 questions
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1a
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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

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1b
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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

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2
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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.

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

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2
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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.

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3a
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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.

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3b
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(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.

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4
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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.

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5
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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.

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6a
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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.

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6b
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Find the area of the shaded region.

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7a
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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.

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7b
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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.

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8
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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.

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9a
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(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]

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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.

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10a
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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.

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10b
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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.

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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

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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.

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2
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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.

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3a
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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.

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3b
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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.

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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.

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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.

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5a
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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.

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5b
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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.

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