Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) demonstrates that differentiation and integration are inverse processes of one another.

The Fundamental Theorem of Calculus (Part I)

If $f(x)$ is continous on $[a,b]$, then the function $g(x)$ defined by $g(x)={\int }_a^{x}f(t)dt,\text{for }x\in [a,b]$ is differentiable on (a,b), and its derivative is $g^{\prime }(x)=f(x)$

If $F(x)$ is an antiderivative of $f(x)$, then every antiderivative of $f(x)$ can be expressed as $F(x)+C$, for some constant $C$.

The Fundamental Theorem of Calculus (Part II)

If $F(x)$ is any antiderivative of $f(x)$, then ${\int }_a^{b}f(t)dt=F(b)-F(a)$