Question

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

 

Answer 1

\[f'\left( x \right) = x - \frac{1}{x^2}\]
\[ f'\left( x \right) = x - x^{- 2} \]
\[\int f'\left( x \right)dx = \int\left( x - x^{- 2} \right)dx\]
\[ f\left( x \right) = \frac{x^2}{2} - \frac{x^{- 2 + 1}}{- 2 + 1} + C\]
\[ = \frac{x^2}{2} + \frac{1}{x} + C\]
\[f\left( 1 \right) = \frac{1}{2} \left( Given \right)\]
\[ \Rightarrow \frac{1^2}{2} + \frac{1}{1} + C = \frac{1}{2}\]
\[ \Rightarrow C = - 1\]
\[ \therefore f\left( x \right) = \frac{x^2}{2} + \frac{1}{x} - 1\]