本题目来源于试卷: Integral Calculus，类别为 IB数学
[问答题]
Consider the functioes8hlno:zs*73i;hv o z0. t52thtj l an9ib1h+ r:vus-.b yypibi f1f. s $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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