Problems

Evaluate:

1. [(i)] \(\int \frac{dx}{(x^2+a^2)^3}\);
2. [(ii)] \(\int \frac{dx}{(x-2)\sqrt{x^2+2x+3}}\);
3. [(iii)] \(\int \frac{\sin x dx}{4\cos x+3\sin x}\);
4. [(iv)] \(\int_0^{\frac{\pi}{2}} e^{ax} \sin x dx\).

Find the length and area of the loop of \(3x^2 = y(1-y)^2\).

Shew that if two points at a distance \(a\) apart are inverted with respect to an origin distant \(e\) and \(f\) from them, the distance between the inverse points is \(\frac{a}{ef}\), if the radius of inversion is unity. Shew that the problem of inverting three points \(A,B,C\) with respect to a point in their plane into three points which are the vertices of an equilateral triangle has two real solutions.

Shew how to obtain a convergent of a continued fraction of the type \(\frac{1}{a_1+}\frac{1}{a_2+}\dots\) when the two convergents immediately preceding are known. Shew that successive convergents are alternately greater and less than the value of the fraction itself and that the differences between them and the fraction steadily decrease.

Give definitions of \(e^x, \sin x\) and \(\cos x\) which are applicable when \(x\) is a complex number and verify that with these definitions \[ \sin(x+y)=\sin x \cos y + \cos x \sin y. \] Prove that \[ x + \frac{x^4}{4!} + \frac{x^7}{7!} + \dots = \frac{1}{3}e^x + \frac{2}{3}e^{-x/2}\sin\left(\frac{x\sqrt{3}}{2}-\frac{\pi}{6}\right). \]

The equation of a conic is \[ x^2+4xy+y^2-2x-6y=0. \] Find the lengths of its semiaxes, and the coordinates of its centre and foci.

Shew that the function \(\sin x + a\sin 3x\) for values of \(x\) from \(0\) to \(\pi\) has no zeros except the terminal ones if \(-\frac{1}{3}\frac{1}{3}\). Indicate roughly the types of the graph of the function in the four cases.

Illustrate the term 'formula of reduction' for an integral. Find formulae for the cases \[ \text{(i) } \int \sin^n x\,dx, \quad \text{(ii) } \int e^{-ax}\sin^n x\,dx. \]

Obtain the dimensions of the quantities (velocity, force, power, etc.) which occur in dynamics in terms of mass, space, time. Shew that 1 watt, which is \(10^7\) C.G.S. units of power, is equal to \(\frac{1}{746}\) horse-power, taking 1 lb. = 453.6 grams, \(g=32.2\) foot second units = 981 centimetre second units.

Defining simple harmonic motion as the projection on a diameter of uniform circular motion, deduce the velocity and acceleration in terms of the period \(\tau\) and the displacement \(x\). If a particle slides on a smooth cycloid whose axis is vertical and vertex at the lowest point, shew that the motion is simple harmonic. Shew also that the projection of the particle on the axis has a simple harmonic motion.