Q.If $f'(x)=4x-5$ and $f(2)=1$, find $f(x)$.

Integration is the reverse (inverse) operation of differentiation. A function $F(x)$ is called an antiderivative (also primitive or Newton-Leibnitz integral) of a function $f(x)$ on an interval $I$ if $F'(x)=f(x)$ for every $x$ in $I$.

Antiderivatives are not unique: if $F(x)$ is one antiderivative of $f(x)$, then so is $F(x)+c$ for any constant $c$, because $\frac{d}{dx}(F(x)+c)=F'(x)=f(x)$. In fact every antiderivative of $f$ on $I$ has this form, so we write

where $c$ is the arbitrary constant of integration, $f(x)$ is the integrand, and $x$ (in $dx$) is the variable of integration. The family $F(x)+c$ represents infinitely many parallel curves. …