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a.
$\int\limits_{0}^{\pi/4} \frac{1 -2 \sin^{2} x}{1+\sin2x}dx$
$=\int\limits_{0}^{\pi/4} \frac{\cos2x}{1+\sin2x}dx$
$=\frac{1}{2}\int\limits_{0}^{\pi/4} \frac{d(1+\sin2x)}{1+\sin2x}dx$
$=\frac{1}{2}\ln|1+\sin2x|\left|\begin{array}{l}\frac{\pi}{4}\\0\end{array}\right.=\frac{\ln2}{2}$
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