JEE Advance - Mathematics (2009 - Paper 2 Offline - No. 6)
Let $$f:R \to R$$ be a continuous function which satisfies $$f(x) = \int\limits_0^x {f(t)dt} $$. Then, the value of $$f(\ln 5)$$ is ____________.
Answer
0
Explanation
We have $$f(x) = \int\limits_0^x {f(t)dt \Rightarrow f(0) = 0} $$
Also, $$f'(x) = f(x),x > 0$$. Therefore, $$f(x) = k,x > 0$$
Hence, $$f(0) = 0$$ and $$f(x)$$ is continuous,
$$f(x) = 0\forall x > 0$$
Since $$f(\ln 5) = 0$$.
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