notes/education/math/MATH1220 (calc II)/Integration by Parts.md

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