A) \[(2,x)\]
B) \[(1,x)\]
C) \[(1,\sqrt{x})\]
D) \[(2,\sqrt{x})\]
Correct Answer: D
Solution :
\[\int{\frac{\sqrt{x}}{x(x+1)}}dx=k{{\tan }^{-1}}m\] Put \[\left\{ \begin{matrix} x={{\tan }^{2}}\theta \\ dx=2\tan \theta .{{\sec }^{2}}\theta d\theta \\ \end{matrix} \right.\] \[=\int{\frac{\tan \theta }{{{\tan }^{2}}\theta .{{\sec }^{2}}\theta }.(2tan\theta .se{{c}^{2}})}d\theta \] \[=2\int{d\theta .=2\theta =2{{\tan }^{-1}}\sqrt{x}}=k{{\tan }^{-1}}(m)\] On comparing, we get \[(k,m)=(2,\sqrt{x})\]
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