The integration by parts formula is:
$$ \int udv = uv - \int vdu $$

## Deriving the Integration by Parts Formula
$$ \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) $$
1. Integrating both sides, we get:
$$\int \frac{d}{dx} (f(x)g(x))dx = \int [f'(x)g(x) + f(x)]$$
2. Through the distributive property of integrals,
$$ = \int f'(x)g(x)dx + \int f(x)g'(x)dx $$
3. Therefore:
$$f(x)g(x) = \intf'(x)g(x)dx $$