Integration by Parts 分部積分法

\[\int u dv = uv - \int v du\]

用於相乘的積分

推導:

\[d \big( u v \big) = u \ d(v) + v \ d(u) \\ \to \int d \big( u v \big) = \int \Big[ u \ d(v) + v \ d(u) \Big] \\ \to u v = \int \Big( u \ d(v) \Big) + \int \Big( v \ d(u) \Big) \\ \to \int u dv = uv - \int v du\]

Example 1

\[\int x \ cosx dx \\ = \int u dv = uv - \int v du \\ = x \ sinx - \int sinx dx \\ = x \ sinx + cosx + C\]

細節:

\[let \ \ u=x, dv = cosx dx \\ du = dx, v = \int cosx d_x = sinx\]

快速的作法:

\[\int x \ cosx dx \\ = \int \underbrace{x}_{u} \ d \underbrace{sinx}_{v} \\ = uv - \int v \ du \\ = x \ sinx - \int sinx dx \\ = x \ sinx + cosx + C\]

Example 2

\[\int \underbrace{\ln x}_{u} \ d \underbrace{x}_{v} \\ = u v - \int v du = x \ln x - \int x d \ln x \\ = x \ln x - \int x x^{-1} dx \\ = x \ln x - \int 1 dx \\ = x \ln x - x + C\]

細節:

\[\frac{d}{dx} \ln x = x^{-1} \\ \to d \ln x = x^{-1} dx\]