If \(F(m, n) = \int_1^\infty (x-1)^m x^{-n} dx\), where \(m\) and \(n\) are positive integers satisfying \(n > m+2\), find relations between \(F(m, n-1)\) and \(F(m,n)\), and between \(F(m-1, n)\) and \(F(m, n)\). Hence find the value of \(F(m, n)\).
Sketch the curve whose equation in polar coordinates is \(r=1+\cos 2\theta\). Prove that the length of the curve corresponding to \(0 \le \theta \le 2\pi\) is \[ 8 + \frac{4}{\sqrt{3}} \log(2+\sqrt{3}). \]
A point \(P\) is situated at a distance \(f\) from the centre of a thin spherical shell of radius \(a\), and \(R\) is the distance of \(P\) from any point of the shell. Show that the mean value of \(R\), averaged with respect to elements of area of the shell, is \(f+(a^2/3f)\) or \(a+(f^2/3a)\) according as \(f\) is greater or less than \(a\).
If \(y=\psi_n(x)\) is a solution of the equation \[ \frac{d^2y}{dx^2} + \frac{2(n+1)}{x} \frac{dy}{dx} + y = 0, \] show that \(Y=\frac{1}{x}\frac{d}{dx}\{\psi_n(x)\}\) satisfies the equation \[ \frac{d^2Y}{dx^2} + \frac{2(n+2)}{x} \frac{dY}{dx} + Y=0. \] Hence show that, if \(n\) is a positive integer, \(\psi_n(x) = \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\sin x}{x}\) satisfies the former differential equation.
If \(y\) is defined as a function of \(x\) by the equation \(f(x,y)=0\), and subscripts denote partial differentiation, show that \[ \frac{d^2y}{dx^2} = -\frac{f_{xx}f_y^2 - 2f_{xy}f_x f_y + f_{yy} f_x^2}{f_y^3}. \] Illustrate the use of this formula by finding the radius of curvature of the ellipse \(x^2+xy+y^2=3\) at the point \((1,1)\).
If \[ p_r = \frac{1 \cdot 4 \cdot 7 \dots (3r-2)}{3 \cdot 6 \cdot 9 \dots (3r)} \quad (r=1, 2, \dots), \] show that \[ 2(p_{2n+1} + p_1 p_{2n} + \dots + p_n p_{n+1}) = \frac{2 \cdot 5 \cdot 8 \dots (6n+2)}{3 \cdot 6 \cdot 9 \dots (6n+3)} \quad (n=0, 1, 2, \dots). \]
If \(k\) and \(l\) are positive numbers, and the sequence \((a_n)\) satisfies the recurrence relation \[ a_{n+1} = k a_n + l a_{n-1}, \] prove that \[ \lim_{n\to\infty} \frac{a_n}{\alpha^n} = \frac{a_2 - \beta a_1}{\alpha(\alpha-\beta)}, \] where \(\alpha\) is the positive root and \(\beta\) the negative root of the equation \[ x^2 - kx - l = 0. \]
If \[ I = \int_0^\pi \frac{x \cos^2 x \sin x}{\sqrt{(1+3 \cos^2 x)}}\,dx, \] show that \[ I = \frac{\pi}{2} \int_0^\pi \frac{\cos^2 x \sin x}{\sqrt{(1+3\cos^2 x)}}\,dx, \] and hence evaluate \(I\).
If \(y = \frac{\sin x}{x}\), show that \[ \frac{d^n y}{dx^n} = u_n \sin x + v_n \cos x, \] where \(u_n\) and \(v_n\) are polynomials in \(1/x\) satisfying the relations \begin{align*} x u_{2n+1} + (2n+1)u_{2n} &= 0, \\ x v_{2n+1} + (2n+1)v_{2n} &= (-1)^n. \end{align*} Prove also that \begin{align*} u_{2n+1} &= u'_{2n} - v_{2n}, \\ v_{2n+1} &= u_{2n} + v'_{2n}, \end{align*} where the dash denotes differentiation with respect to \(x\). Deduce that \(u=u_{2n}\) satisfies the equation \[ x^2 \frac{d^2u}{dx^2} + 2(2n+1)x \frac{du}{dx} + [x^2+2n(2n+1)]u = (-1)^n x. \]
Find the equation of the normal at the point \(T(ct, c/t)\) to the rectangular hyperbola \(xy=c^2\). The normals at three points \(P, Q, R\), with parameters \(p, q, r\), are concurrent. Prove that \[ qr+rp+pq+up+uq+ur=0, \] where \[ pqru = -1. \] Find the quadratic equation whose roots give the feet of the two further normals from the point of intersection of the normals at the points whose parameters have given values \(\theta, \phi\).