A) \[\frac{{{x}^{a+1}}}{a+1}+c\]
B) \[\frac{{{x}^{a+1}}}{a+1}+\frac{{{a}^{x}}}{\log a}+c\]
C) \[{{x}^{a+1}}+{{a}^{x}}+c\]
D) \[\frac{{{x}^{a+1}}}{a-1}+\frac{\log a}{{{a}^{x}}}+c\]
E) \[\frac{{{x}^{a+1}}}{a+1}-\frac{{{a}^{x}}}{\log a}+c\]
Correct Answer: B
Solution :
\[\int{({{e}^{a\log x}}+{{e}^{x\log a}})}dx\] \[=\int{({{e}^{\log {{x}^{a}}}}+{{e}^{\log {{a}^{x}}}})}dx\] \[=\int{({{x}^{a}}+{{a}^{x}})}\,dx=\frac{{{x}^{a+1}}}{a+1}+\frac{{{a}^{x}}}{{{\log }_{e}}a}+c\]
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