Integration by Parts

Theorem
Let $f$ and $g$ be real functions which are continuous on the closed interval $\left[{a. . b}\right]$.

Let $f$ and $g$ have primitives $F$ and $G$ respectively on $\left[{a. . b}\right]$.

Then:
 * $\displaystyle \int_a^b f \left({t}\right) G \left({t}\right) \mathrm dt = \left[{F \left({t}\right) G \left({t}\right)}\right]_a^b - \int_a^b F \left({t}\right) g \left({t}\right)\mathrm dt$

This is frequently written as:
 * $\displaystyle \int u \ \mathrm dv = u v - \int v \ \mathrm du$

where it is understood that $u, v$ are functions of the independent variable.

Proof
By Product Rule for Derivatives, we have $D \left({FG}\right) = f G + F g$.

Thus $FG$ is a primitive of $f G + F g$ on $\left[{a. . b}\right]$.

Hence, by the Fundamental Theorem of Calculus:
 * $\displaystyle \int_a^b \left({f \left({t}\right) G \left({t}\right) + F \left({t}\right) g \left({t}\right)}\right) \mathrm dt = \left[{F \left({t}\right) G \left({t}\right)}\right]_a^b$

The result follows.