Calculus

Calculus Level 4

Let f : R → R f: \mathbb R \to \mathbb R be a continuous function satisfying f ( x ) = ∫ 0 x f ( t ) d t \displaystyle f(x) = \int_0^x f(t) \, dt . Find the value of f ( ln ⁡ 5 ) f(\ln5) .

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Anish J by anish j , 22, India

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1 solution

Tom Engelsman
Jun 13, 2017

If we differentiate the above equation with respect to x x , then we arrive at the ODE f ′ ( x ) = f ( x ) f'(x) = f(x) with the initial condition f ( 0 ) = 0. f(0) = 0. Solving this ODE produces f ( x ) = C e x f(x) = Ce^{x} and solving for the real constant C C gives:

0 = C ⋅ e 0 = C ( 1 ) ⇒ C = 0 0 = C \cdot e^{0} = C(1) \Rightarrow C = 0

hence, f ( x ) = 0 f(x) = 0 for all x ∈ R x \in \mathbb{R} and f ( l n ( 5 ) ) = 0 . f(ln(5)) = \boxed{0}.

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