Question

Let $f:\left[1,\infty \right]\to \left[2,\infty \right]$ if $f\left(1\right)=2.$ be differentiable function such that $6{\int }_1^{x}f\left(t\right)dt=3xf\left(x\right)-x^3$ then the value of $f\left(2\right)$ is....

Answer 1

$6{\int }_1^{2}f\left(t\right)dt=3xf\left(x\right)-x^3$

Diff. both sides.

$6f\left(x\right)=3f\left(x\right)+3xf\left(x\right)-3x^2$

$y=\frac{xdy}{dx}-x^2$

$\frac{dy}{dx}-\frac{y}{x}=x$

$I.E=e^{\int \frac{1}{x}dx}=\frac{1}{x}$

$\frac{y}{x}=\int \frac{1}{x}\times xdx$

$y=x^2+C$

$f\left(1\right)=2$ $C=1$

$y=x^2+1$

$f\left(2\right)=2^2+1=5$