📚 Integration by Parts | 分部积分法
Integration by parts is a core integration technique in A-Level Edexcel Mathematics. It allows you to integrate products of functions by transforming a challenging integral into a simpler one. Based on the product rule for differentiation, this method is essential for tackling a wide range of integrals, from polynomials multiplied by exponentials to logarithmic and inverse trigonometric functions. A solid grasp of how to choose u and dv, along with recognising when to apply the rule repeatedly or handle cyclic cases, is vital for exam success.
分部积分法是A-Level Edexcel数学中一种核心积分技巧。它能将函数的乘积积分转化为更简单的积分，其基础是乘法的微分法则。掌握该方法对于处理从多项式乘指数函数到对数函数和反三角函数的各类积分至关重要。扎实理解如何选择 u 和 dv，并识别何时重复应用或应对循环情况，是考试成功的关键。
1. The Formula: Derivation and Statement | 公式推导与表述
The integration by parts formula is derived directly from the product rule for differentiation. For two differentiable functions u(x) and v(x), the derivative of their product is (uv)′ = u′v + uv′. Rearranging gives uv′ = (uv)′ − u′v. Integrating both sides with respect to x yields ∫ uv′ dx = uv − ∫ u′ form as ∫ u dv = uv − ∫ v du. This is the fundamental identity used throughout this topic.
分部积分公式直接由乘积法则推导而来。对于两个可导函数 u(x) 和 v(x)，其乘积的导数为 (uv)′ = u′v + uv′。整理得 uv′ = (uv)′ − u′v。两边对 x 积分即得 ∫ uv′ dx = uv − ∫ u′v dx。令 u = u(x)，v = v(x)，可写成微分形式 ∫ u dv = uv − ∫ v du。这是贯穿本主题的基础恒等式。
∫ u dv = u v − ∫ v du
2. Choosing u and dv: The LIATE Rule | 选择 u 和 dv：LIATE 规则
The effectiveness of integration by parts hinges on a sensible choice of u and dv. The widely used LIATE rule gives a priority order for setting u: Logarithmic functions (e.g., ln x), Inverse trigonometric functions (e.g., arctan x), Algebraic functions (e.g., xⁿ), Trigonometric functions (e.g., sin x), Exponential functions (e.g., eˣ , becomes dv. This strategy almost always makes ∫ v du simpler than the original integral.
分部积分的效果取决于 u 和 dv 的合理选择。广泛使用的 LIATE 规则给出了设定 u 的优先顺序：L 对数函数、I 反三角函数、A 代数函数、T 三角函数、E 指数函数。选择被积函数中最早出现的类别作为 u，剩余部分连同 dx 作为 dv。这一策略几乎总能令 ∫ v du 比原积分更简单。
3. Basic Polynomial Times Exponential | 基本多项式乘指数函数
Example: Evaluate ∫ x eˣ dx. Here x belongs to the Algebraic category (A) and eˣ to Exponential (E). By LIATE, set u = x and dv = eˣ dx. Then du = dx and v = eˣ. Applying the formula gives ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C. Notice how the remaining integral of eˣ is straightforward to compute.
例题：计算 ∫ x eˣ dx。此处 x 属于代数函数类别 (A)，eˣ 属于指数函数 (E)。按 LIATE 规则，设 u = x，dv = eˣ dx。于是 du = dx，v = eˣ。代入公式得 ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C。注意剩下的 eˣ 积分十分简单。
4. Integrating Products with Trigonometric Functions | 包含三角函数的乘积积分
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