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. An integral cancels out an antiderivative, therefore:

f(x)g(x) = \int f'(x)g(x)dx + \int f(x)g'(x)dx

4. Moving terms around:

\int f(x)g'(x)dx = f(x)g(x) - \int f'(x)g(x)dx

Now, let u = f(x) and v = g(x), then dv = g'(x)dx and du = f'(x)dx.

Examples