If \( \int \sin ^{-1}\left(\sqrt{\frac{x}{1+x}}\right) d x=A(x) \cd...

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If \( \int \sin ^{-1}\left(\sqrt{\frac{x}{1+x}}\right) d x=A(x) \cdot \tan ^{-1} \sqrt{x}+B(x)+c \),
\( \mathrm{P}^{142} \)
W where \( c \) is a constant of integration, then the ordered pair \( (A(x), B(x)) \) can be:
(1) \( (x+1,-\sqrt{x}) \)
(2) \( (x+1, \sqrt{x}) \)
(3) \( (x-1,-\sqrt{x}) \)
(4) \( (x-1, \sqrt{x}) \)
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