Prerequisites

### Integration by parts

For ordinary functions of one variable, the rule for integration by parts follows immediately from integrating the product rule \begin{eqnarray*} \frac{d}{dx}(fg) &=& \frac{df}{dx}g+f\frac{dg}{dx}\\ \int_a^b\frac{d}{dx}(fg)\, dx &=& \int_a^b\frac{df}{dx}g\, dx + \int_a^b f\frac{dg}{dx}\, dx\\ \left. fg \right\vert_a^b &=& \int_a^b\frac{df}{dx}g\, dx + \int_a^b f\frac{dg}{dx}\, dx \end{eqnarray*} Rearranging, we obtain $$\int_a^b\frac{df}{dx}g\, dx = \left. fg \right\vert_a^b - \int_a^b f\frac{dg}{dx}\, dx$$