Given that y=secθy = \sec \thetay=secθ
Express yyy in terms of cosθ\cos \thetacosθ.
Hence, show that dydθ=secθtanθ \frac{dy}{d\theta} = \sec \theta \tan \thetadθdy=secθtanθ
Show that for 0<θ<π20 < \theta < \frac{\pi}{2}0<θ<2π, y2−1y=sinθ \frac{\sqrt{y^2-1}}{y} = \sin \thetayy2−1=sinθ
Use the substitution x=3secux = 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∫x2x2−91dx can be written as k∫cosu du k \int \cos u \, duk∫cosudu where kkk is a constant to be found.
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∫x2x2−91dx=9xx2−9+C
Practise Edexcel A Level Maths 11.5 Integration by Substitution with exam-style questions for A Level Maths. 59 questions, matched to the Edexcel A Level Maths (9MA0) 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.