Let for some function $\mathrm{y}=f(x), \int_0^x t f(t) d t=x^2 f(x), x \gt 0$ and $f(2)=3$. Then $f(6)$ is equal to

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Aicharya · Accepted Answer

$\begin{aligned} & \int_0^x t f(t) d t=x^2 f(x) \ & \Rightarrow \quad x f(x)=2 x f(x)+x^2 f^{\prime}(x) \ & \Rightarrow x^2 \cdot f^{\prime}(x)=-x f(x) \ & \Rightarrow \quad \frac{f^{\prime}(x)}{f(x)}=-\frac{1}{x} \ & \ & \quad \int \frac{f^{\prime}(x)}{f(x) d x}=-\int \frac{1}{x} d x \ & \Rightarrow \quad \ln f(x)=-\ln x+C \ & \ & \quad f(2)=3 \Rightarrow \ln 3=-\ln 2+C \ & \quad C=\ln 6 \ & \Rightarrow \quad f(x)=\frac{6}{x} \Rightarrow f(6)=1\end{aligned}$

Let for some function $\mathrm{y}=f(x), \int_0^x t f(t) d t=x^2 f(x), x \gt 0$ and $f(2)=3$. Then $f(6)$ is…

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