Chọn phương án B.

\(\begin{aligned}
\displaystyle\int\limits_0^1 \dfrac{2abx+a+b}{(1+ax)(1+bx)}\mathrm{\,d}x&=\displaystyle\int\limits_0^1 \dfrac{(abx+a)+(abx+b)}{(1+ax)(1+bx)}\mathrm{\,d}x\\
&=\displaystyle\int\limits_0^1 \dfrac{a(bx+1)+b(ax+1)}{(1+ax)(1+bx)}\mathrm{\,d}x\\
&=\int\limits_0^1\left(\dfrac{a}{ax+1}+\dfrac{b}{bx+1}\right )\mathrm{\,d}x\\
&=\ln|ax+1|\bigg|_0^1+\ln|bx+1|\bigg|_0^1\\
&=\ln|(a+1)(b+1)|=0.\end{aligned}\)

Suy ra \(|(a+1)(b+1)|=1\)
\(\Leftrightarrow|ab+a+b+1|=1\)
\(\Leftrightarrow\left[\begin{array}{l}S=ab+a+b+1=1\\ S=ab+a+b+1=-1\end{array}\right.\)
\(\Leftrightarrow\left[\begin{array}{l}S=ab+a+b=0\\ S=ab+a+b=-2.\end{array}\right.\)