Question

If f' (x) = x − $\frac{1}{x^2}$ and f(1) = $\frac{1}{2},$ find f(x)

Answer 1

$$\begin{gathered} f\text{'}\left(x\right)=x-\frac{1}{x^2} \\ f\text{'}\left(x\right)=x-x^{-2} \\ \int f\text{'}\left(x\right)dx=\int \left(x-x^{-2}\right)dx \\ f\left(x\right)=\frac{x^2}{2}-\frac{x^{-2+1}}{-2+1}+\mathrm{C} \\ =\frac{x^2}{2}+\frac{1}{x}+\mathrm{C} \\ f\left(1\right)=\frac{1}{2}\left(\mathrm{Given}\right) \\ \Rightarrow \frac{1^2}{2}+\frac{1}{1}+\mathrm{C}=\frac{1}{2} \\ \Rightarrow \mathrm{C}=-1 \\ \therefore f\left(x\right)=\frac{x^2}{2}+\frac{1}{x}-1 \end{gathered}$$