Fundamental theorem of calculus : The fundamental theorem of calculus states that $\frac{d}{dx}\int \limits_{0}^{x}f(t)dt$ $=f(x).$ In other words the function $F$ defined by $F(x)=\int \limits_{0}^{x}f(t)dt$ is an antiderivative of $f.$ As the difference of two antiderivatives is a constant, this can equivalently be stated by saying that if $F$ is an antiderivative of $f$, then $\int \limits_{a}^{b}f(t)dt$ $=F(b)-F(a).$ The fundamental theorem of calculus provides a link between differentiation and integration and a simple way of calculating integrals via the computation of an antiderivative. For example as $\frac{1}{2}x^2$ is an antiderivative of $x$ we have $\int \limits_{a}^{b}tdt$ $=\frac{1}{2}b^2-\frac{1}{2}a^2.$