A) \[{{\sin }^{-1}}\left( \frac{x}{a} \right)+c\]
B) \[{{\log }_{e}}|x+\sqrt{{{x}^{2}}-{{a}^{2}}}|+c\]
C) \[{{\log }_{e}}|x-\sqrt{{{x}^{2}}-{{a}^{2}}}|+c\]
D) \[\frac{x\sqrt{{{x}^{2}}-{{a}^{2}}}}{2+c}\]
Correct Answer: B
Solution :
\[I=\int_{{}}^{{}}{\frac{dx}{\sqrt{{{x}^{2}}-{{a}^{2}}}}}\]. Put\[x=a\sec \theta \Rightarrow dx=a\sec \theta \,.\,\tan \theta \,d\theta \] \[\therefore \,\,\,I=\int_{{}}^{{}}{\frac{a\sec \theta \,.\,\tan \theta \,d\theta }{a\tan \theta }}=\int_{{}}^{{}}{\sec \theta \,d\theta }\] \[=\log (\sec \theta +\tan \theta )+=\log \left( \frac{x}{a}+\frac{\sqrt{{{x}^{2}}-{{a}^{2}}}}{a} \right)+c\] \[=\log (x+\sqrt{{{x}^{2}}-{{a}^{2}}})+c\].
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