Evaluate the integrals:
\[ \int_0^\infty \frac{x^{p-1}}{1+x+x^2}\,dx \quad (0
Explain how to distinguish the two "sides" of a bilateral surface. Define \(\iint f(x,y,z)dydz\) taken over a specified side of a given bilateral surface and show how to calculate it in terms of \(u\) and \(v\) when the equations to the surface are \[ x=\theta(u,v), \quad y=\phi(u,v), \quad z=\psi(u,v). \] Find the value of \[ \iint x^3y^3z^5dydz \] taken over the outer side of the octant of the surface of the ellipsoid \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \] for which \(x,y,z\) are all positive.
The Green's function \(G(x,y,z)\) associated with a given closed surface \(S\) and origin \((a,b,c)\) in its interior is defined by the properties:
Develop the Lagrange-Charpit method of solving a partial differential equation and illustrate by considering \[ p^2+q^2-2px-2qy+1=0. \]
Prove that two elliptic functions with the same periods, zeros and poles have a constant ratio. Assuming that the functions sn \(u\), cn \(u\), dn \(u\) have the usual periods, zeros and poles, apply this method to compare \[ \text{cn } u \text{ cn }(u+v) - \text{cn } v, \quad \text{sn } u \text{ sn }(u+v), \] proving that \[ \text{cn } u \text{ cn }(u+v) + \text{sn } u \text{ dn } v \text{ sn }(u+v) - \text{cn } v = 0. \] Deduce addition theorems for the functions sn, cn.
Find the condition that the equations \[ ax^2+2bx+c=0, \quad a'x^2+2b'x+c'=0 \] may have a common root. Prove that, if \(a, c, a', b', c'\) are given so that \(b'^2 > a'c'\), two real values \(b_1, b_2\) of \(b\) can be found to ensure a common root; and form the equation whose roots are the other roots of the equations \[ ax^2+2b_1x+c=0, \quad a'x^2+2b_2x+c'=0. \]
Prove that, if \(A-x^2=u\), (\(x>0, u>0\)), then \(\sqrt{A}\) lies between \(x\) and \(x+u/2x\). Hence prove that, if \(P = N^2+\frac{2}{3}n\) and \(Q = N^2+\frac{1}{3}n\), the difference between \((N^2+n)^{1/2}\) and \(N\sqrt{P/Q}\) is less than \(\frac{5}{36}n^3N^{-1}P^{-1}Q^{-1}\) when \(N\) and \(n\) are positive. Apply the method to express \(\sqrt{53}\) as a decimal, estimating the degree of accuracy obtained.
Prove that, if two infinite series of positive terms \(\sum u_n, \sum v_n\) are such that \(u_n/v_n\) tends to a finite limiting value, not zero, when \(n\) tends to infinity, the series are both convergent or both divergent. Deduce a rule for the convergence or divergence of \(\sum u_n\), when \(u_{n+1}/u_n\) tends to a limit \(k\). Examine for different positive or negative values of \(z\) and \(p\) the convergence or divergence of the series whose \(n\)th terms are \[ \text{(i) } z^n, \quad \text{(ii) } \frac{\sqrt{n+1}-\sqrt{n}}{n^p}. \]
The side \(BC\) of a triangle \(ABC\) is divided at \(D\) so that \(BD:DC=m:n\), where \(m+n=1\). Prove that, if \(R, R_1, R_2\) are the radii of the circles \(ADB, ADC, ABC\), then \[ bR_1 = cR_2 = R(mb^2+nc^2-mna^2)^{1/2}. \] Verify the results obtained in the limiting case when \(m\) tends to zero.
In order to obtain the height \(z\) of an aeroplane above the horizontal plane of a triangle \(ABC\) its angular altitude \(\alpha\) is observed at \(C\), and simultaneously at \(A, B\) are observed its bearings \(\theta, \phi\) measured in the horizontal plane from \(AB\) and \(BA\) respectively towards \(C\). Prove that \[ z^2\cot^2\alpha\sin(\theta+\phi) = b^2\sin\theta\cos\phi+a^2\sin\phi\cos\theta - 2ab\sin C\sin\theta\sin\phi - z^2\sin\theta\sin\phi\cot(\theta+\phi). \]