A) \[\sin {{a}^{x}}+c\]
B) \[{{a}^{x}}\sin {{a}^{x}}+c\]
C) \[\frac{1}{{{(\log a)}^{2}}}\sin {{a}^{x}}+c\]
D) \[\log \sin {{a}^{x}}+c\]
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{\frac{1}{\log a}({{a}^{x}}\cos {{a}^{x}})\,dx}\] Put \[{{a}^{x}}=t\Rightarrow {{a}^{x}}dx=\frac{dt}{\log a},\] then it reduces to \[\int_{{}}^{{}}{\frac{1}{{{(\log a)}^{2}}}\cos t\,dt}=\frac{1}{{{(\log a)}^{2}}}\sin t+c=\frac{1}{{{(\log a)}^{2}}}\sin {{a}^{x}}+c\].
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