Selecting Techniques for Antidifferentiation
Introduced| English | Meaning |
|---|---|
| antidifferentiation/ˌæntɪˌdɪfəˌrenʃɪˈeɪʃn/ | Finding a function whose derivative is the given function, with an arbitrary additive constant on an interval. |
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.
Read the integrand, pick the method
y = ax³ + bx
Whether an integrand is a plain power, a composite, or an improper fraction decides which antiderivative tool to use.
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.
What is the fastest method for $\int (3x^2+e^x)\,dx$?
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Both terms are standard antiderivatives → basic rules.
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.
Which integral is best done by u-substitution?
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A composite with its inner derivative ($u=x^2+1$, $du=2x\,dx$) → u-sub.
Select all correct method signals.
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u-sub is not universal — match the method to the structure.
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.
The first step for $\int\dfrac{x^2}{x-1}\,dx$ is...
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Numerator degree ≥ denominator degree → divide first.
You can check any antiderivative by differentiating it back to the integrand.
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Differentiating your answer should return $f$.
Order the techniques from first to last to try.
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Basic rule, then u-sub, then rewrite.
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.
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).
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.