Mathematics

Let \(f(x) = \frac{ax}{x^2 + 1}\), \(-8 \leq x \leq 8\), \(a \in \mathbb{R}\). The graph of \(f\) is shown below.

The region between \(x = 3\) and \(x = 7\) is shaded.

1. Show that \(f(-x) = -f(x)\).[2]
2. Given that \(f'(x) = \frac{2ax(x^2 - 3)}{(x^2 + 1)^3}\), find the coordinates of all points of inflexion.[7]
3. It is given that \(\displaystyle\int f(x) \mathrm{d}x = \frac{a}{2} \ln{(x^2 + 1)} + C\).
   1. Find the area of the shaded region, giving your answer in the form \(p \ln{q}\).
   2. Find the value of \(\displaystyle \int_{4}^{8} 2f(x-1) \mathrm{d}x\).[7]