Selecting Techniques for Antidifferentiation · 选择原函数求解技术
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| antidifferentiation/ˌæntɪˌdɪfəˌrenʃɪˈeɪʃn/ | 求原函数 | qiú yuán hán shù |
Which antiderivative tool fits?
- You now have several ways to integrate — the skill is choosing the right technique fast.
- Antidifferentiation 求原函数 has no single algorithm, so you read the integrand's structure.
- A quick triage points you to a basic rule, a substitution, or an algebraic rewrite.
- This lesson is the decision guide that ties the integration methods together.
该用哪样求原函数的工具?
- 你现在有好几种积分方法——技巧是快速选对技术。
- 求原函数没有单一算法,所以你要读被积函数的结构。
- 快速分诊会指向一条基本规则、一次换元、或一次代数改写。
- 这一课是把积分方法串起来的决策指南。
Read the integrand, pick the method · 观察被积函数，选择方法
y = ax³ + bx
Whether an integrand is a plain power, a composite, or an improper fraction decides which antiderivative tool to use. · 被积函数是简单幂函数、复合函数还是不定分式，决定了使用哪种原函数工具。
First: does a basic rule fit directly?
- Is the integrand a power, or one of the standard forms ($\sin, \cos, e^x, \tfrac1x$)?
- If so, apply the basic rule and add $+C$ — no tricks needed.
- $\displaystyle\int (x^3+\cos x)\,dx$ is just term-by-term basic antiderivatives.
- Always try this first; it's the fastest path.
首先:基本规则能直接套吗?
- 被积函数是一个幂,或标准形式之一($\sin, \cos, e^x, \tfrac1x$)吗?
- 若是,套用基本规则并加 $+C$——不用技巧。
- $\displaystyle\int (x^3+\cos x)\,dx$ 就是逐项的基本原函数。
- 永远先试这个;它最快。
What is the fastest method for $\int (3x^2+e^x)\,dx$? · 对于$\int (3x^2+e^x)\,dx$，最快的方法是？
Both terms are standard antiderivatives → basic rules. · 两项均为标准原函数 → 使用基本规则。
Next: is there an inner function with its derivative?
- Spot a composite with its inner derivative present → u-substitution.
- $\displaystyle\int 2x\,e^{x^2}\,dx$: inner $u=x^2$, and $2x\,dx=du$ is right there.
- If $du$ is off by a constant, adjust; if the inner derivative is truly absent, u-sub won't help.
- This handles the large class of chain-rule-shaped integrals.
其次:有内层函数及其导数吗?
- 发现一个复合函数且其内层导数在场 → 换元积分。
- $\displaystyle\int 2x\,e^{x^2}\,dx$:内层 $u=x^2$,而 $2x\,dx=du$ 就在那里。
- 若 $du$ 相差一个常数,调整;若内层导数确实不在,换元帮不上忙。
- 这处理一大类链式法则形状的积分。
Which integral is best done by u-substitution? · 哪个积分最适合用u代换？
A composite with its inner derivative ($u=x^2+1$, $du=2x\,dx$) → u-sub. · 含内层导数的复合函数（$u=x^2+1$, $du=2x\,dx$）→ u代换。
Select all · 所有 correct method signals. · 选择所有正确的方法信号。
u-sub is not universal — match the method to the structure. · u代换并非万能——需根据结构匹配方法。
Otherwise: rewrite the algebra first
- No basic rule and no clean substitution? Rewrite the integrand.
- Improper rational fraction → long division; irreducible quadratic denominator → complete the square.
- Expand a product, split a fraction into separate terms, or simplify — then re-triage.
- After the rewrite, a basic rule or u-sub usually finishes it.
否则:先改写代数
- 没有基本规则,也没有干净的换元?改写被积函数。
- 假有理分式 → 长除法;不可约二次分母 → 配方。
- 展开一个乘积、把分式拆成几项、或化简——然后重新分诊。
- 改写之后,一条基本规则或换元通常就完成它。
The first step for $\int\dfrac{x^2}{x-1}\,dx$ is... · 对于$\int\dfrac{x^2}{x-1}\,dx$，第一步是...
Numerator degree ≥ denominator degree → divide first. · 分子次数 ≥ 分母次数 → 先除。
You can check any antiderivative by differentiating it back to the integrand. · 你可以通过对原函数求导来验证其是否为被积函数。
Differentiating your answer should return $f$. · 对你的答案求导应返回$f$。
Order the techniques from first to last to try. · 按尝试顺序排列各项技术。
Basic rule, then u-sub, then rewrite. · 先基本规则，再u代换，最后改写。
Don't reach for a heavy technique before checking the easy ones. Try a basic rule first, then u-substitution, and only then an algebraic rewrite. And confirm every antiderivative by differentiating it back to the integrand — that catches most mistakes on the spot.
别在检查简单方法之前就动用重技术。先试基本规则,再换元积分,最后才是代数改写。并通过把每个原函数求导回被积函数来确认——那能当场发现大多数错误。
Choose a method for each, then note the first step:
- $\displaystyle\int (3x^2 + e^x)\,dx$: basic rules → $x^3+e^x+C$.
- $\displaystyle\int x\,\sqrt{x^2+1}\,dx$: u-sub ($u=x^2+1$, $du=2x\,dx$).
- $\displaystyle\int \dfrac{x^2}{x-1}\,dx$: long division first (improper fraction).
先给每个选方法,再指出第一步:
- $\displaystyle\int (3x^2 + e^x)\,dx$:基本规则 → $x^3+e^x+C$。
- $\displaystyle\int x\,\sqrt{x^2+1}\,dx$:换元($u=x^2+1$,$du=2x\,dx$)。
- $\displaystyle\int \dfrac{x^2}{x-1}\,dx$:先长除法(假分式)。
Selecting an antidifferentiation technique: try a basic rule first; if the integrand is a composite with its inner derivative, use u-substitution; otherwise rewrite the algebra (long division / completing the square / expanding) and re-triage. Always verify by differentiating your answer.
选择求原函数技术:先试基本规则;若被积函数是含内层导数的复合函数,用换元积分;否则改写代数(长除法 / 配方 / 展开)再重新分诊。永远通过对答案求导来验证。