A) \[\log (3+8{{e}^{x}})+C\]
B) \[\frac{1}{4}\log (3+8{{e}^{x}})+C\]
C) \[\frac{1}{2}\log (3+8{{e}^{x}})+C\]
D) none of these
Correct Answer: B
Solution :
Since, f'(x)=f(x), therefore, \[f(x)=\alpha {{e}^{x}}\] Since, f(0) = 2, therefore, \[f(x)=2{{e}^{x}}\] \[\therefore \]\[I=2\int_{{}}^{{}}{\frac{{{e}^{x}}}{3+8{{e}^{x}}}}dx\]Put\[{{e}^{x}}=t,\therefore dt={{e}^{x}}dx\] \[\therefore \]\[I=2\int_{{}}^{{}}{\frac{dt}{3+8t}\Rightarrow I=\frac{1}{4}\log (3+8{{e}^{x}})+C}\]
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