Find formulae of reduction for \[ \int \sin^n x\,dx, \quad \int x(1+x^2)^n\,dx, \] where \(n\) is positive. If \[ f(m,n) = \int_0^{\frac{\pi}{2}} \cos^m x \cos nx\,dx, \] prove that \[ (m+n)f(m,n)=mf(m-1,n-1). \]
Find the area of a loop of the curve \[ (x^2+4y^2)^2 = x^2-9y^2. \]
If \begin{align*} a(x+y+b)+x^2y^2+bxy(x+y) &= 0, \\ a(z+x+b)+z^2x^2+bzx(z+x) &= 0, \end{align*} and \(y\) and \(z\) are unequal, prove that \[ a(y+z+b)+y^2z^2+byz(y+z)=0. \]
Prove that if \(0
Prove that the number of ways in which \(n\) different letters can be arranged in a row is \(n!\). Prove also that when they are arranged in a row in any way they can be divided up into \(r\) words in \(^{n-1}C_{r-1}\) ways, so that the total number of sentences of \(r\) words and \(n\) letters which can be formed from them is \[ n!(n-1)!/(n-r)!(r-1)!. \]
A coin is to be tossed twice; what is the chance that heads will turn up at least once? Point out the error in the following solution, given by D'Alembert: "Only three different events are possible; (i) heads the first time, which makes it unnecessary to toss again; (ii) tails the first time and heads the second, (iii) tails both times. Of these three events two are favourable; therefore the required chance is \(\frac{2}{3}\)."
Solve the equations:
Show how to determine \(u_n\) from the equation \[ Au_n+Bu_{n+1}+Cu_{n+2}=0, \] where \(A, B, C\) are constants, when the values of \(u_0\) and \(u_1\) are given; and show that the complete solution of \[ u_n - 2u_{n+1}\cos\theta+u_{n+2}=0 \] is of the form \(H\cos n\theta+K\sin n\theta\).
If \(AB, BC, CD\) are three sides of a quadrilateral of lengths \(a,b,c\) respectively, and if \(\angle ABC=\theta, \angle BCD=\phi\), and the angle between \(AB\) and \(DC\) produced is \(\psi\), prove that \[ AD^2 = a^2+b^2+c^2 - 2ab\cos\theta - 2bc\cos\phi - 2ca\cos\psi. \]
If \(\alpha+\beta+\gamma+\delta=2\pi\), show that \[ (\sin 2\alpha+\sin 2\beta+\sin 2\gamma+\sin 2\delta)^2 = (\cos\beta.\cos\gamma-\cos\alpha.\cos\delta)(\cos\gamma.\cos\alpha-\cos\beta.\cos\delta)(\cos\alpha.\cos\beta-\cos\gamma.\cos\delta). \] % Note: there appears to be a factor missing from the original scan. This is a common identity for the area of a cyclic quadrilateral, but it's usually written with a factor of 16 or 64. I will transcribe as written.