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.

$\blacksquare$

Notes

The technique of solving an integral in the form $\displaystyle \int_a^b f \left({t}\right) G \left({t}\right) \mathrm dt$ in this manner is called integration by parts.

Its validity as a solution technique stems from the fact that it may be possible to choose $f$ and $G$ such that $G$ is easier to differentiate than to integrate.

Thus the plan is to reduce the integration to one such that $\displaystyle \int_a^b F \left({t}\right) g \left({t}\right) \mathrm dt$ is easier to solve than $\displaystyle \int_a^b f \left({t}\right) G \left({t}\right) \mathrm dt$

It may be, of course, that a further application of this technique is needed before the solution can be extracted.

Sources

• Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $14.4$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 13.21$
• For a video presentation of the contents of this page, visit the Khan Academy.