A) \[\log ({{x}^{e}}+{{e}^{x}})+c\]
B) \[e\log ({{x}^{e}}+{{e}^{x}})+c\]
C) \[\frac{1}{e}\log ({{x}^{e}}+{{e}^{x}})+c\]
D) None of these
Correct Answer: C
Solution :
Put \[{{x}^{e}}+{{e}^{x}}=t\Rightarrow e({{x}^{e-1}}+{{e}^{x-1}})\,dx=dt,\] \[\int_{{}}^{{}}{\frac{{{x}^{e-1}}+{{e}^{x-1}}}{{{x}^{e}}+{{e}^{x}}}}\,dx=\frac{1}{e}\int_{{}}^{{}}{\frac{dt}{t}}=\frac{1}{e}\log t=\frac{1}{e}\log ({{x}^{e}}+{{e}^{x}})+c\].
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