Question

If $f^{\prime }\left(x\right)=x-\frac{1}{x^2}$ and $f\left(1\right)=\frac{1}{2}$ then find $f\left(x\right)$.

Answer 1

Given,

$f^{\prime }\left(x\right)=x-\frac{1}{x^2}$

Integrating both sides we get,

$f\left(x\right)=\frac{x^2}{2}+\frac{1}{x}+c$ [ Where $c$ is integrating constant]

Given $f\left(1\right)=\frac{1}{2}$.

or, $\frac{1}{2}=\frac{1}{2}+1+c$

or, $c=-1$

Using value of $c$ we have,

$f\left(x\right)=\frac{x^2}{2}+\frac{1}{x}-1$.