Mathematics · Definite Integration

JEE Main 2025 — 28 January, Morning Shift — Question 16

Let for some function $y=f(x), \int_{0}^{x} t f(t) d t=x^{2} f(x)$ , $x>0$ and $f(2)=3$ . Then $f(6)$ is equal to :

Answer: A

$\int_{0}^{\mathrm{x}} \mathrm{tf}(\mathrm{t}) \mathrm{dt}=\mathrm{x}^{2}.f(\mathrm{x}), \mathrm{x}>0$

Diff. both side w.r. to x

$x f(x)=x^{2} f^{\prime}(x)+2 x f(x)$

$-x f(x)=x^{2} f^{\prime}(x)$

$\int \frac{f^{\prime}(x)}{f(x)} d x=\int \frac{-1}{x} d x$

$\log f(x)=-\log x+\log c$

$\mathrm{f}(\mathrm{x})=\frac{\mathrm{c}}{\mathrm{x}}$

$\mathrm{f}(2)=3 \Rightarrow 3=\frac{\mathrm{c}}{2} \Rightarrow \mathrm{c}=6$

$f(x)=\frac{6}{x}$

$\mathrm{f}(6)=1$

Answer key and solution verified before publishing.

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