Q.Given $f'(x) = x - \dfrac{3}{x^3}$ and $f(1) = \dfrac{11}{2}$, find $f(x)$.

An indefinite integral is the reverse operation to differentiation. If a function $g(x)$ has derivative $f(x)$, i.e. $g'(x)=f(x)$, then $g(x)$ is called a primitive (or antiderivative) of $f(x)$, and we write $\int f(x)\,dx=g(x)+c$. Because the derivative of any constant is zero, $g(x)+c$ is a primitive of $f(x)$ for every real number $c$ — so an indefinite integral does not represent one function but an entire one-parameter family of curves, each a vertical translate of the others, all sharing the same tangent-slope function $f(x)$ at every $x$. Geometrically, at any fixed $x$-value the tangents to every curve in the family are parallel, even though the curves pass …