A) \[-2\]
B) 3
C) \[-1\]
D) 0
Correct Answer: D
Solution :
given that \[f(x)=\int_{0}^{x}{f(t)}dt\] |
Clearly \[f(0)=0.\] |
Also\[f'(x)=f(x)\] \[\Rightarrow \]\[\frac{f'(x)}{f(x)}=1\] |
Integrating both sides with respect to x, we get |
\[\int{\frac{f'(x)}{f(x)}dx}=\int{1\,\,dx}\] \[\Rightarrow \]In \[f(x)=x+\ln C\Rightarrow f(x)=C{{e}^{x}}\] |
Now \[f(0)=0\Rightarrow C{{e}^{x}}=0\Rightarrow C=0\] |
\[\therefore \]\[f(x)=0\,\forall \,x\Rightarrow f(\ln 5)=0\] |
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