Shew how to integrate \[ \frac{1}{(x-x_0)\sqrt{(ax^2+2bx+c)}}, \] and prove that the integral will be algebraical if and only if \(ax_0^2+2bx_0+c=0\).
A shot whose mass is \(m\) penetrates to a depth \(a\) when fired at a plate of mass \(M\) which is free to move. Determine the depth to which it would have penetrated had the plate been fixed.
Trace the curve \[ y = x \pm \sqrt{\{x(x-1)(2-x)\}}. \]
Circles are drawn with their centres on the circle \(x^2+y^2=1\) and touching the axis of \(y\). Shew that the axis of \(y\) and the curve \[ 4(x^2+y^2-1)^2 = 27x^2 \] form the envelope of the system of circles; and trace the curve.
A heavy particle slides down a smooth vertical circle of radius \(R\) from rest at the highest point. Shew that on leaving the circle it moves in a parabola whose latus rectum is \(\frac{16}{27}R\).
Differentiate \(\cos x\) from first principles. Differentiate \[ \sin^{-1}\left[\frac{2\sqrt{\{(\alpha-x)(x-\beta)\}}}{\alpha-\beta}\right]. \]
Under the action of constant tractive effort \(P\) by the engine, a train of total mass \(m\) starting from rest at \(A\) attains its maximum speed \(V\); the pull of the engine is then reduced so that for a time the speed is maintained at its value \(V\), after which the steam is shut off and the brakes applied, bringing the train to rest at the point \(B\). The distance \(AB\) is \(l\), the time of run between \(A\) and \(B\) is \(\frac{4l}{3V}\), the rail resistance is \(\frac{3}{5}\frac{mV^2}{l}\), and the brake resistance is \(\frac{12}{5}\frac{mV^2}{l}\), both these being independent of the speed. Prove that \[ P = \frac{18}{5}\frac{mV^3}{l}. \]
Show that the function \[ y = ax^2 + 2bx + c + A \cos mx + B \sin mx \] satisfies an equation of the form \[ \frac{d^4y}{dx^4} + k^2 \frac{d^2y}{dx^2} = 0, \] determining the value of \(k\).
Calculate \[ \int (x \cos x)^2 dx, \quad \int x \log x dx, \quad \int_0^\pi \frac{dx}{13+5 \cos x}. \]
Show that the surface generated by the revolution of the cardioid \[ r = a(1-\cos\theta) \] about the line \(\theta=0\) is \(\displaystyle\frac{32}{5}\pi a^2\).