A) \[{{\tan }^{-1}}\sqrt{x}\]
B) \[{{\tan }^{-1}}\left( \frac{\sqrt{x}}{3} \right)\]
C) \[\frac{2}{3}{{\tan }^{-1}}\sqrt{x}\]
D) \[\frac{2}{3}{{\tan }^{-1}}\left( \frac{\sqrt{x}}{3} \right)\]
Correct Answer: D
Solution :
We have, \[I=\int_{{}}^{{}}{\frac{dx}{\sqrt{x}(x+9)}}\] Put \[\sqrt{x}=t\], squaring both sides, we get \[x={{t}^{2}}\] and \[dx=2tdt\] \[\therefore \]\[I=2\int_{{}}^{{}}{\frac{dt}{{{t}^{2}}+{{3}^{2}}}}=\frac{2}{3}{{\tan }^{-1}}\left( \frac{t}{3} \right)\] Þ \[I=\frac{2}{3}{{\tan }^{-1}}\left( \frac{\sqrt{x}}{3} \right)\].
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