Integrate \[ \int \frac{dx}{1+e^{2x}}, \quad \int \frac{d\theta}{\sin^2\theta\cos^2(\theta+\alpha)}, \] and shew that if \(a\) and \(b\) are both positive and \(a>b\), \[ \int_0^\pi \frac{d\theta}{a+b\cos\theta} = \frac{\pi}{\sqrt{a^2-b^2}}. \]
Sketch the curve given by the equation \[ y^2 = \frac{x^2(3a-x)}{a+x}. \] Shew that the coordinates of any point on the curve may be taken as \((a\sin 3\theta\csc\theta, a\sin 3\theta\sec\theta)\), and prove that the area of the loop of the curve is equal to the area between the curve and its asymptote.