IGCSEFurther MathsEdexcelExam QuestionsCalculusIntegration

Integration (Edexcel IGCSE Further Pure Maths): Exam Questions

Exam code: 4PM1

22 mins5 questions
1
8 marks
Graph showing a curve crossing the x-axis at points (-1,0), O, (b,0), and (a,0) with shaded area between (-1,0) and (b,0). Diagram not accurately drawn.

Figure 3 shows a sketch of the curve with equation y space equals space straight f left parenthesis x right parenthesis, which passes through the points with coordinates space left parenthesis negative 1 comma space 0 right parenthesis comma space left parenthesis b comma space 0 right parenthesis and left parenthesis a comma space 0 right parenthesis where 0 space less than space b space less than space a.

Given that straight f apostrophe left parenthesis x right parenthesis space equals space 6 x squared space – space 26 x space plus space 12 find,

(i) the value of a,

(ii) the value of b.

How did you do?

2
2 marks

Find integral 4 sin space 5 theta space sin 3 theta space straight d theta

How did you do?

3
4 marks

use algebraic integration to find the exact value of

integral subscript 0 superscript straight pi over 6 end superscript open parentheses 4 space cos cubed space theta space minus space sin space 2 theta close parentheses space straight d theta

How did you do?

4
4 marks

Using a formula, show that

cos space 2 theta equals 2 space cos squared space theta minus 1

Hence show that

integral subscript pi over 3 end subscript superscript fraction numerator 3 pi over denominator 4 end fraction end superscript open parentheses 2 space cos to the power of 2 space end exponent theta minus 1 close parentheses   d theta equals negative fraction numerator a plus square root of b over denominator c end fraction

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

Graph with two curves, \(C_1\) and \(C_2\), intersecting x-axis at O. Shaded region R between points A and B on horizontal axis \(θ\). Diagram not to scale.

Figure 3 shows part of the curve C subscript 1 with equation y equals 2 space cos squared space theta minus 1and part of the curve C subscript 2 with equation y equals negative cos space theta

Point B is the intersection of C subscript 1 and C subscript 2 as shown in Figure 3

Point Aopen parentheses fraction numerator 3 straight pi over denominator 4 end fraction comma 0 close parentheses is the intersection of C subscript 1 with the theta-axis as shown in Figure 3

Point E open parentheses pi over 2 comma 0 close parentheses is the intersection of C subscript 2 with the theta-axis as shown in Figure 3

How did you do?

5
4 marks

Using calculus, find the exact value of

integral subscript 0 superscript pi over 2 end superscript open parentheses fraction numerator 4 space tan space theta over denominator 1 plus tan squared space theta end fraction minus cos space 5 theta plus 2 close parentheses   straight d theta

How did you do?