本题目来源于试卷: Differential Calculus，类别为 IB数学
[问答题]
Consider the functiocq(a4k4 wrz52vy mfke5rbx 7/nsp.d4.0/lt: i x axftky $f(x)=\sin x,-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ and $g(x)=\frac{2 x \sqrt{1-x^{2}}}{1-2 x^{2}}$, $x \in \mathbb{R}$, x$ \neq \pm \frac{1}{\sqrt{2}}$ .
1. Find an expression for $(g \circ f)(x)$ , stating its domain.
2. Hence show that $(g \circ f)(x)=\tan 2 x$ .
3. Letting $y=(g \circ f)(x)$ , find an exact value for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $ x=\frac{\pi}{3}$ .
4. Show that the area bounded by the graph of $y=(g \circ f)(x)$ , the x -axis and the lines x=0 and $x=\frac{\pi}{3}$ is $\frac{1}{2} \ln 2$ .
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