Integration by Parts
分部积分法
Integration by parts is a useful strategy for simplifying some integrals. It is based on the combination rule for differentiation and the general approach can be summarized by:
分部积分法是一种简化某些积分的有用策略。它基于微分的乘积法则，其一般方法可以概括为：
This technique is particularly appropriate for removing a linear term multiplying an exponential. For example, the integral
这种技术特别适用于消除指数函数前的线性项。例如，积分
can be simplified by the identification
可以通过识别来简化
Then u can be differentiated and dv can be integrated.
然后可以对 u 进行求导，对 dv 进行积分。
The value of the integral is then
积分的值则为
| Applications
应用 |
|
If a higher power term of x multiplies the exponential, then the process of integration by parts can be repeated to reduce the power term. Two stages are necessary to integrate a square term.
如果x的高次幂项乘以指数函数，则可以通过分部积分法重复进行，以降低幂次。要积分一个平方项需要两个阶段。
| Example:
例： |
|
Three stages are necessary if the integrand has a cubic term, and so on.
如果被积函数中含有三次项，那么需要分三个阶段来处理。
| Example:
例： |
|
Integration concepts
索引积分概念
| HyperPhysics*****HyperMath*****Calculus
HyperPhysics ***** HyperMath ***** 微积分 | R Nave |
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