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1.11 H: Integration

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Question 356

Given that

y=sec⁡θ y = \sec \theta y=secθ
ai.

Express yyy in terms of cos⁡θ\cos \thetacosθ.

[1]
aii.

Hence, show that

dydθ=sec⁡θtan⁡θ\frac{dy}{d\theta} = \sec \theta \tan \theta dθdy​=secθtanθ
[2]
aiii.

Show that for 0<θ<π20 < \theta < \frac{\pi}{2}0<θ<2π​,

y2−1y=sin⁡θ\frac{\sqrt{y^2-1}}{y} = \sin \theta yy2−1​​=sinθ
[2]
bi.

Use the substitution x=3sec⁡ux = 3 \sec ux=3secu to show that for x>3x > 3x>3, the integral

∫1x2x2−9 dx\int \frac{1}{x^2 \sqrt{x^2 - 9}} \, dx ∫x2x2−9​1​dx

can be written as

k∫cos⁡u duk \int \cos u \, du k∫cosudu

where kkk is a constant to be found.

[3]
bii.

Hence, show that

∫1x2x2−9 dx=x2−99x+C\int \frac{1}{x^2 \sqrt{x^2 - 9}} \, dx = \frac{\sqrt{x^2 - 9}}{9x} + C ∫x2x2−9​1​dx=9xx2−9​​+C
[2]

1.11 H: Integration Questions

  1. A Level
  2. /Maths
  3. /1.11 H: Integration

Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.

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