If \( \int \frac{\sin 2x}{(1+\sin x)(2+\sin x)} \, dx = a \log |1+\sin x| - b \log |2+\sin x| + c \), then the value of \( a \) and \( b \) is __________.
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For integrals involving \( \sin x \) and \( \cos x dx \), use \( t=\sin x \) and then apply partial fractions.