UV′=(UV)′−U′V UV'=(UV)'-U'VUV′=(UV)′−U′V
∫UV′dx=UV−∫U′Vdx \int UV' dx= UV - \int U'V dx ∫UV′dx=UV−∫U′Vdx
−−−−−−−−−−−−−−−−−−−−−−−−−− -------------------------- −−−−−−−−−−−−−−−−−−−−−−−−−−
↓ \downarrow ↓
观察模式
UU′V′V U \qquad\qquad U' \\ V' \qquad \qquad VUU′V′V
eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
呃呃呃呃呃呃呃呃呃呃呃呃呃呃呃呃呃呃呃呃 呃呃呃呃呃呃呃呃 呃呃呃呃呃呃呃呃 呃呃
使用积分法，能更容易发现积分中的重复部分，例如：
∫0+∞e−xsinxdx \int_{0}^{ +\infty } e^{-x}sinx dx∫0+∞e−xsinxdx
注意：是-x,不是x
当然使用复变函数看似更加容易一点：
∫0+∞e−xsinxdx=Im∫0+∞e−xeixdx=Im∫0+∞e(−1+i)xdx=Im1−1+ie(1+i)x∣0+∞ \qquad \int_{0}^{ +\infty } e^{-x}sinx dx \\=Im\int_{0}^{ +\infty } e^{-x}e^{ix}dx \\
=Im\int_{0}^{ +\infty } e^{(-1+i)x}dx \\ =Im \frac { 1 } { -1+i}e^{(1+i)x} |_{0}^{ +\infty } ∫0+∞e−xsinxdx=Im∫0+∞e−xeixdx=Im∫0+∞e(−1+i)xdx=Im−1+i1e(1+i)x∣0+∞
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