This equation shows integration by parts, a powerful technique for integration.
| \( u \) | This is a generic function, that could be of any variable. |
| \( v \) | This is a generic function, that could be of any variable. |
| \( \int \) | This is the symbol for an integral, sometimes referred to as an antiderivative. Graphically, it can be understood as the area between a curve and the axis the integral is taken with respect to. |
| \( \:d \) | This is the symbol for a differential. It represents an infinitesimally small (infinitely close to zero) change in whatever variable it is with respect to. |
Consider the product rule for differentiation:
\[(\htmlClass{sdt-0000000008}{u} \cdot \htmlClass{sdt-0000000019}{v})\htmlClass{sdt-0000000025}{'} = \htmlClass{sdt-0000000008}{u}\htmlClass{sdt-0000000025}{'} \cdot \htmlClass{sdt-0000000019}{v} + \htmlClass{sdt-0000000019}{v}\htmlClass{sdt-0000000025}{'} \cdot \htmlClass{sdt-0000000008}{u}\]
We will assume that both \( \htmlClass{sdt-0000000008}{u} \) and \( \htmlClass{sdt-0000000019}{v} \) are functions of \( \htmlClass{sdt-0000000003}{x} \).
We can now integrate both sides with respect to \( \htmlClass{sdt-0000000003}{x} \) to get:
\[ \htmlClass{sdt-0000000060}{\int} (\htmlClass{sdt-0000000008}{u} \cdot \htmlClass{sdt-0000000019}{v})\htmlClass{sdt-0000000025}{'} \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} = \htmlClass{sdt-0000000060}{\int} (\htmlClass{sdt-0000000008}{u} \htmlClass{sdt-0000000025}{'} \cdot \htmlClass{sdt-0000000019}{v} + \htmlClass{sdt-0000000019}{v} \htmlClass{sdt-0000000025}{'} \cdot \htmlClass{sdt-0000000008}{u}) \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} \]
More coming soon. The rest of the steps essentially just finish the integral.
Coming soon...