A) \[2{{a}^{\sqrt{x}}}{{\log }_{e}}a+c\]
B) \[2{{a}^{\sqrt{x}}}{{\log }_{a}}e+c\]
C) \[2{{a}^{\sqrt{x}}}{{\log }_{10}}a+c\]
D) \[2{{a}^{\sqrt{x}}}{{\log }_{a}}10+c\]
Correct Answer: B
Solution :
Put \[\sqrt{x}=t\Rightarrow \frac{1}{2}\frac{1}{\sqrt{x}}\,dx=dt,\] then \[\int_{{}}^{{}}{\frac{{{a}^{\sqrt{x}}}}{\sqrt{x}}\,dx}=2\int_{{}}^{{}}{{{a}^{t}}dt}=\frac{2{{a}^{t}}}{{{\log }_{e}}a}+c=2{{a}^{\sqrt{x}}}{{\log }_{a}}e+c.\]
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