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JEE Main 202528 JanuaryMorning Shift
Exam
JEE Main
Year
2025
Date
28 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Definite Integration

JEE Main 2025 · 28 January · Morning Shift · Mathematics · Definite Integration

Let for some function y=f(x), ∫_0^x t f(t) d t=x^2 f(x), x 0 and...

Let for some function y=f(x),∫0xtf(t)dt=x2f(x),x>0\mathrm{y}=f(x), \int_0^x t f(t) d t=x^2 f(x), x\gt0 and f(2)=3f(2)=3. Then f(6)f(6) is equal to

Options

  1. A1
  2. B3
  3. C6
  4. D2

Correct answer

A. 1

Step-by-step solution

∫0xtf(t)dt=x2f(x)⇒xf(x)=2xf(x)+x2f′(x)⇒x2⋅f′(x)=−xf(x)⇒f′(x)f(x)=−1x∫f′(x)f(x)dx=−∫1xdx⇒ln⁡f(x)=−ln⁡x+Cf(2)=3⇒ln⁡3=−ln⁡2+CC=ln⁡6⇒f(x)=6x⇒f(6)=1\begin{aligned} & \int_0^x t f(t) d t=x^2 f(x) \\ & \Rightarrow \quad x f(x)=2 x f(x)+x^2 f^{\prime}(x) \\ & \Rightarrow x^2 \cdot f^{\prime}(x)=-x f(x) \\ & \Rightarrow \quad \frac{f^{\prime}(x)}{f(x)}=-\frac{1}{x} \\ & \\ & \quad \int \frac{f^{\prime}(x)}{f(x) d x}=-\int \frac{1}{x} d x \\ & \Rightarrow \quad \ln f(x)=-\ln x+C \\ & \\ & \quad f(2)=3 \Rightarrow \ln 3=-\ln 2+C \\ & \quad C=\ln 6 \\ & \Rightarrow \quad f(x)=\frac{6}{x} \Rightarrow f(6)=1\end{aligned}
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