Let f:[1, ∞) → R be a differentiable function such that f(1)=1/3 and 3 ∫₁ˣ f(t) d t=x f(x)-(x³)/3, x [1, ∞). Let e denote the base of the natural logarithm. Then the value of f(e) is

Integral Calculus: JEE Advanced 2023 Mathematics question

JEE Advanced 2023 · JEE Adv 2023 Paper 2, Paper 2 · Q1

JEE Advanced 2023JEE Adv 2023 Paper 2, Paper 2MathematicsHardIntegral Calculus369× since 2002

Let

f:[1, \infty) \rightarrow \mathbb{R}

be a differentiable function such that

f(1)=\frac{1}{3}

and

3 \int_{1}^{x} f(t) d t=x f(x)-\frac{x^{3}}{3}, x \in[1, \infty)

. Let

e

denote the base of the natural logarithm. Then the value of

f(e)

JEE Advanced 2023 Integral Calculus Question with Answer (JEE Adv 2023 Paper 2 Paper 2, Q1) | JEE Arc

JEE Advanced 2023 · JEE Adv 2023 Paper 2, Paper 2 · Q1

JEE Advanced 2023JEE Adv 2023 Paper 2, Paper 2MathematicsHardIntegral Calculus369× since 2002

Let

f:[1, \infty) \rightarrow \mathbb{R}

be a differentiable function such that

f(1)=\frac{1}{3}

and

3 \int_{1}^{x} f(t) d t=x f(x)-\frac{x^{3}}{3}, x \in[1, \infty)

. Let

e

denote the base of the natural logarithm. Then the value of

f(e)