Integration by Parts | 分部积分法

📚 Integration by Parts | 分部积分法

Integration by parts is one of the most powerful techniques in the Edexcel A-Level Pure Mathematics specification. It is used to integrate products such as x sin x and x eˣ, and also single functions whose derivatives are simpler, such as ln x and arctan x.

分部积分法是 Edexcel A-Level 纯数学考纲中最有力的技巧之一。它用于求 x sin x、x eˣ 等乘积的积分，也用于求导后变得更简单的单个函数，如 ln x 和 arctan x 的积分。

The technique reverses the product rule for differentiation, so mastering it allows you to handle a wide range of integrals that cannot be solved by direct reverse differentiation.

该技巧逆转了乘法求导法则，因此掌握它可以帮助你解决大量无法直接反求导的积分。

1. The Formula and Derivation | 公式与推导

The integration by parts formula is usually written as:

分部积分公式通常写作：

∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx

Here u and v are functions of x. In words, the integral of u times dv/dx equals u times v minus the integral of v times du/dx.

其中 u 和 v 都是 x 的函数。也就是说，u 乘以 dv/dx 的积分等于 u 乘以 v 减去 v 乘以 du/dx 的积分。

The formula comes from the product rule: d/dx(uv) = u′v + uv′. Rearranging gives uv′ = d/dx(uv) − u′v, and integrating both sides produces the formula above.

该公式来自乘法求导法则：d/dx(uv) = u′v + uv′。移项得 uv′ = d/dx(uv) − u′v，两边积分即可得到上述公式。

The aim is to replace a difficult integral with a simpler one by choosing u to become simpler after differentiation and dv/dx to be easy to integrate.

其目的是通过选择合适的 u 使它在求导后变简单，并选择容易积分的 dv/dx，从而用一个更简单的积分替代原来的积分。

2. Choosing u and dv/dx | 选择 u 与 dv/dx

The hardest part is often deciding which factor should be u. A common priority list is LIATE:

最难的部分通常是决定哪个

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