Integration by Parts/Definite Integral

Theorem
Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$.

Let $f$ and $g$ have primitives $F$ and $G$ respectively on $\closedint a b$.

Then:
 * $\displaystyle \int_a^b \map f t \map G t \rd t = \bigintlimits {\map F t \map G t} a b - \int_a^b \map F t \map g t \rd t$

Proof
By Product Rule for Derivatives:
 * $\map D {F G} = f G + F g$

Thus $F G$ is a primitive of $f G + F g$ on $\closedint a b$.

Hence, by the Fundamental Theorem of Calculus:
 * $\displaystyle \int_a^b \paren {\map f t \map G t + \map F t \map g t} \rd t = \bigintlimits {\map F t \map G t} a b$

The result follows.