Solve the equations
Find the coefficient of \(x^n\) in the expansion of \((2+3x+x^2)^{-1}\) in ascending powers of \(x\). Prove that, if \(n\) is a positive integer, the coefficient of \(x^n\) in \((1+x+x^2)^n\) is \[ 1 + \frac{n(n-1)}{(1!)^2} + \frac{n(n-1)(n-2)(n-3)}{(2!)^2} + \dots. \]
Sum the series to \(n\) terms
If \(n\) is a prime number prove that \(a^n-a\) is divisible by \(n\). If \(n\) is a prime number of the form \(6m+1\), prove that \(3^n-2^n-1=M(294n)\).
Differentiate \(\sin^{-1}(\csc\theta\sqrt{\cos 2\theta})\), \(\tan^{-1}\{x/(1+\sqrt{1+x^2})\}\). Find the \(n\)th differential coefficient of \(x\tan^{-1}x\).
Prove that \(x=\pi/3\) will make \(\cos^{-1}(a\sin x)+2\cos^{-1}(a\cos\frac{x}{2})\) a minimum if \(0
If \(x=r\cos\theta\), \(y=r\sin\theta\), find \(\dfrac{\partial x}{\partial r}, \dfrac{\partial\theta}{\partial y}\), and give a geometrical interpretation. Prove that \[ \left(\frac{\partial^2\theta}{\partial x \partial y}\right)^2 - \frac{\partial^2\theta}{\partial x^2}\frac{\partial^2\theta}{\partial y^2} = \frac{1}{r^4}. \]
If \(\phi\) is the angle between the radius vector and the tangent of a curve, prove that \[ \tan\phi = r\frac{d\theta}{dr}. \] Find \(\phi\) at a point of the curve \(r^2=a^2\cos 2\theta\). If the normal at the point \((r,\theta)\) cuts the initial line in \(G\), prove that \(OG = 2ar(a^2+r^2)/(a^4+2r^4)\) where \(O\) is the origin.
Interpret the expressions (i) \(\int x \frac{dy}{ds} ds\), (ii) \(\int y \frac{dx}{ds} ds\), (iii) \(\int (x \frac{dy}{ds} - y \frac{dx}{ds}) ds\), where the integrals are taken round the perimeter of a closed curve. Find the area of a loop of the curve \(x^5+y^5=ax^2y^2\).
Find the integrals \[ \int \frac{dx}{\sqrt{2+x+x^2}}, \quad \int \frac{dx}{1+x^3}, \quad \int_{\alpha}^{\pi-\alpha} \frac{dx}{\tan x - \tan\alpha}, \] \[ \int_0^\pi \frac{dx}{(3+2\cos x)^2}, \quad \int_0^1 x^{n-1}(1-x)^n dx. \]