Substitution, parts and partial fractions
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| integration by parts/ˌɪntɪˈɡreɪʃn baɪ pɑːts/ | 分部积分法 | fēn bù jī fēn fǎ |
Can we integrate the two factors separately?
- A simple-looking product such as xe^x cannot be integrated by separately integrating each factor.
- This lesson studies integration by parts 分部积分法: An integration method based on the derivative of a product.
能否分别对两个因子进行积分？
- 形式简单的乘积（如 xe^x）不能通过对每个因子单独积分来求解。
- 本课研究分部积分法：一种基于乘积求导法则的积分方法。
Choose the mathematical structure
- Use substitution when an inner derivative appears as a factor. By parts, integral u v prime =uv-integral u prime v. For a rational function, divide first if needed, then decompose into partial fractions.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
选择数学结构
- 当内部函数的导数作为因子出现时使用换元法。分部积分公式为 ∫u v' = uv - ∫u' v。对于有理函数，如有必要先做多项式除法，再分解为部分分式。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
Which description correctly defines integration by parts?
An integration method based on the derivative of a product.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For integral xe^x dx, take u=x and v prime=e^x. Then the integral is xe^x-e^x+C. For integral 2x/(x²+1) dx, substitute w=x²+1, dw=2x dx; the answer is ln(x²+1)+C.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
对于 ∫xe^x dx，取 u=x 且 v'=e^x。则积分结果为 xe^x-e^x+C。对于 ∫2x/(x²+1) dx，令 w=x²+1，dw=2x dx；答案为 ln(x²+1)+C。
Substitution, parts and partial fractions
V/π = integral from 0 to 3 of x² dx = 9
The widget shows the volume integrand y²=x² for y=x; compare the exact value 9 with the volume check.
Evaluate the integral of xe^x from 0 to 1.
An antiderivative is (x-1)e^x. At 1 and 0 it gives 0 and -1, so the integral is 1.
Test a tempting shortcut
- Choosing u and v prime well matters: the remaining integral should become simpler. In a definite substitution, either change the limits or return to x before applying the original limits.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The integral of a product equals the product of its separate integrals. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 合理选择 u 和 v' 至关重要：剩余积分应变得更简单。在定积分换元时，要么更换积分限，要么在代入原积分限前先还原为 x。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
积的积分等于各自积分的乘积。此说法错误。请解释它违反了哪一定义或假设。
Find the coefficient of ln(x²+1) in the integral of 2x/(x²+1).
Substitution w=x²+1 gives dw=2x dx, so the coefficient of ln(x²+1) is 1.
The integral of a product equals the product of its separate integrals.
Choosing u and v prime well matters: the remaining integral should become simpler. In a definite substitution, either change the limits or return to x before applying the original limits.
Interpret a new situation
- Check by differentiating. For volumes of revolution around the x-axis use V=π integral y² dx; do not confuse the square of a function with the integral of the function.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
解读新情境
- 通过求导进行验证。绕x轴旋转体的体积使用 V=π∫y² dx；切勿将函数的平方与函数的积分混淆。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
Find the volume divided by π when y=x is revolved about the x-axis from 0 to 3.
V/π=integral x² dx from 0 to 3=[x³/3]=9.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL further mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
An integration method based on the derivative of a product. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- Edexcel IAL 进阶数学；官方单元 FP3。其他单元的补充内容在范围审查中已列出；不属于额外的加分要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
一种基于积的导数的积分方法。请选择关系式，展示该方法，检查其适用条件并解释结果。