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  4. If ∫ sin -1(√(x/1+x)) dx=A(x) tan -1(√x)+B(x)+C, where C is a constant of integration, then the ordered pair ( A (x ), B ( x )) can be :

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Q. 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 $( A (x ), B ( x ))$ can be :

JEE MainJEE Main 2020Integrals

Solution:

Put $ x=\tan ^{2} \theta$
$ \Rightarrow dx =2 \,\tan \,\theta \,\sec ^{2} \theta \,d \theta$
$\int \theta .\left(2 \tan \theta \cdot \sec ^{2} \theta\right) d \theta$
$\downarrow \,\,\,\,\,\,\,\downarrow$
$ I\,\,\,\,\,\, II $ (By parts)
$=\theta \cdot \tan ^{2} \theta-\int \tan ^{2}\, \theta d \,\theta$
$=\theta \cdot \tan ^{2} \theta-\int\left(\sec ^{2} \theta-1\right) d \theta$
$=\theta\left(1+\tan ^{2} \theta\right)-\tan \theta+ C$
$=\tan ^{-1}(\sqrt{ x })(1+ x )-\sqrt{ x }+ C$