A researcher models the radial gradient of a potential field V V\,V using the equation dVdr=1r2r2−16\displaystyle \frac{dV}{dr} = \frac{1}{r^2 \sqrt{r^2 - 16}}drdV=r2r2−161 for r>4r > 4r>4. To solve for VVV, the researcher considers the transformation y=secϕy = \sec \phiy=secϕ.
(i) Express y y\,y in terms of cosϕ\cos \phicosϕ.
(ii) Hence, show that dydϕ=secϕtanϕ\displaystyle \frac{dy}{d\phi} = \sec \phi \tan \phidϕdy=secϕtanϕ.
(iii) Show that for 0<ϕ<π2\displaystyle 0 < \phi < \frac{\pi}{2}0<ϕ<2π, y2−1y=sinϕ\displaystyle \frac{\sqrt{y^2-1}}{y} = \sin \phiyy2−1=sinϕ.
(i) Use the substitution r=4secϕr = 4 \sec \phir=4secϕ to show that for r>4r > 4r>4, the integral ∫1r2r2−16 dr\displaystyle \int \frac{1}{r^2 \sqrt{r^2 - 16}} \, dr∫r2r2−161dr can be written as k∫cosϕ dϕ k \int \cos \phi \, d\phi\,k∫cosϕdϕ where k k\,k is a constant to be determined.
(ii) Hence, show that ∫1r2r2−16 dr=r2−1616r+C\displaystyle \int \frac{1}{r^2 \sqrt{r^2 - 16}} \, dr = \frac{\sqrt{r^2 - 16}}{16r} + C∫r2r2−161dr=16rr2−16+C.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, 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.
