Evaluate the integrals \[ \int\frac{dx}{(x^2+a^2)^2}, \quad \int\frac{dx}{(x^2-1)\sqrt{x^2+x-1}}, \quad \int_0^{\frac{\pi}{2}}\frac{dx}{\sqrt{2x+1}-\sqrt{x+2}}, \quad \int_0^{\frac{\pi}{2}}\sin^{2n}xdx. \] If \[ u_n = \int_0^{\frac{\pi}{m}} x^n\sin mxdx, \] prove that \[ u_n = \frac{n\pi^{n-1}}{m^2 2^{n-1}} - \frac{n(n-1)}{m^2}u_{n-2} \] if \(m\) is an integer of the form \(4r+1\).
Trace the curve \[ y^2(1+x^2)-4y+1=0, \] and find its area.