CourseCourseConceptConceptDocsDocuments

Edexcel A-Level Mathematics A2 P3.5 Integration Questions

Practise algebraic integration of standard functions, recognised derivatives and definite integrals in exact forms and area contexts.

Syllabus
First assessment 2019
Course
Mathematics YMA01
Level
A2

Exam points

  • integrate standard powers, trig and exponential functions before applying exact limits
  • use algebraic integration to produce forms involving ln, π or surds
  • find total area by splitting regions where a curve changes sign

Covered syllabus points

P3.5.1 - 1P3.5.2 - Integration by recognition

Question 1

[Maximum number: 4]
g(x)=2x2−5x+8x−2g(x)=\frac{2 x^{2}-5 x+8}{x-2}

Hence use algebraic integration to show that

∫48 g(x)dx=α+βln⁡3\int_{4}^{8} \mathrm{~g}(x) \mathrm{d} x=\alpha+\beta \ln 3

where α\alpha and β\beta are integers to be found.

Question 2

[Maximum number: 4]

Hence, using algebraic integration, find the exact value of

∫π12π8(5−4cos⁡23x)dx\int_{\frac{\pi}{12}}^{\frac{\pi}{8}}\left(5-4 \cos ^{2} 3 x\right) \mathrm{d} x

Question 3

[Maximum number: 5]

Figure 4 shows a sketch of part of the curve with equation

y=(1+2cos⁡2x)2y=(1+2\cos2x)^2
Figure 4

Figure 4

The curve touches the positive x-axis for the second time when x=a, as shown in Figure 4.
The regions bounded by the curve, the y-axis and the x-axis up to x=a are shown shaded in Figure 4.

Find, using algebraic integration and making your method clear, the exact total area of the shaded regions. Write your answer in simplest form.

All question bank results loaded