# If $\int \sin ^{-1}\left ( \sqrt{\frac{x}{1+x}} \right )dx=A(x)\tan ^{-1}(\sqrt{x})+B(x)+C,$ where $C$ is a constant of integration, then the ordered pair $\inline (A(x),\; B(x))$ can be : Option: 1   Option: 2 Option: 3 Option: 4

$\int \sin ^{-1}\left ( \sqrt{\frac{x}{1+x}} \right )dx=A(x)\tan ^{-1}(\sqrt{x})+B(x)+C,$

$\begin{array}{l} \text { Put } x=\tan ^{2} \theta \Rightarrow d x=2 \tan \theta \sec ^{2} \theta d \theta \\ \int \theta \cdot\left(2 \tan \theta \cdot \sec ^{2} \theta\right) d \theta \end{array}$

Now do the integration by parts

$\\\mathrm{\theta\rightarrow 1st\;function}\\\mathrm{2\tan\theta\sec^2\theta\rightarrow 2nd\;function}$

$\begin{array}{l} =\theta \cdot \tan ^{2} \theta-\int \tan ^{2} \theta \mathrm{d} \theta \\ =\theta \cdot \tan ^{2} \theta-\int\left(\sec ^{2} \theta-1\right) \mathrm{d} \theta \\ =\theta\left(1+\tan ^{2} \theta\right)-\tan \theta+\mathrm{C} \\ =\tan ^{-1}(\sqrt{\mathrm{x}})(1+\mathrm{x})-\sqrt{\mathrm{x}}+\mathrm{C} \end{array}$

$A(x)=1+x,\;\;B(x)=-\sqrt x$

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