Question

How is integration related to anti-derivative?

Answer 1

Relationship between integration with anti-derivative

1. An antiderivative is a differentiable function $\mathrm{F}$ whose derivative is equal to $\mathrm{f}$ (i.e., $\mathrm{F}\text{'}=\mathrm{f}$).
2. Anti-derivative of a function $\mathrm{f}$ is a function which on differentiation yields $\mathrm{f}$.
3. The process of solving for antiderivatives is called anti-differentiation, and its opposite operation is called differentiation, which is the process of finding a derivative.
4. Integration is connected with differentiation through the fundamental theorem of calculus: if $\mathrm{f}$ is a continuous real-valued function defined on a closed interval $\left[a,b\right]$ then, once an antiderivative $\mathrm{F}$ of $\mathrm{f}$ is known, the definite integral of $\mathrm{f}$ over that interval is given by${\int }_a^b\mathrm{f}\left(x\right)\mathrm{d}x=\mathrm{F}\left(b\right)-\mathrm{F}\left(a\right)$.
5. The relationship between integration and anti-derivative is that integration of a function is equal to the sum of anti-derivative of the function and a constant (in case of indefinite integration) or it is the difference of the anti-derivative evaluated at the bounds of integration (in case of definite integration).