AP Calculus AB/BC · Unit 6 · Topic 6.14
Selecting Techniques for Antidifferentiation
Choose basic rules, substitution, division, completing the square, parts, or partial fractions from structure.
1. Topic Focus
Interpret definite integrals as accumulated change, connect sums to integrals, apply both Fundamental Theorems, and select antiderivative techniques.
This topic: Choose basic rules, substitution, division, completing the square, parts, or partial fractions from structure.
2. Key Relationship
Read every symbol with its domain, direction, units, and hypotheses before applying the relationship.
3. Visual Connection
4. Worked Example
A product of a polynomial and e^x suggests parts; g′(x)f(g(x)) suggests substitution.
Write the governing relationship first, carry out the algebra cleanly, and finish with a sentence that answers the mathematical question.
5. Concept Development
1. Selection begins with classification
Antidifferentiation is not a search for the most advanced method. First classify the integrand by structure: a sum of standard forms, a composition, a product, a rational expression, a quadratic form, or an improper integral. The visible structure suggests the first useful move.
2. Inspect the entire problem before integrating
Read the bounds, locate domain restrictions, and identify whether the requested result is indefinite or definite. Infinite bounds and vertical blowups require limits; finite bounds may make a change of variables more efficient; an indefinite answer requires \(+C\).
3. Simplify before choosing a named technique
Expand a small product, split a fraction, cancel only on a domain where cancellation is valid, use a trigonometric identity, or perform polynomial division when it exposes standard antiderivatives. Algebra is often the shortest integration technique.
4. Try basic antiderivative rules first
Also recognize standard exponential, trigonometric, and inverse-trigonometric forms. Do not use substitution merely to rename a variable when a direct rule already applies.
5. Reverse the chain rule with substitution
Look for an inner expression \(g(x)\) and a constant multiple of its derivative:
The match may appear after factoring out a constant or rewriting a numerator. A product alone does not imply integration by parts.
6. Change definite-integral bounds consistently
After substituting \(u=g(x)\), either change both bounds to \(u\)-values and stay in \(u\), or return to \(x\) before applying the original bounds. Mixing \(u\)-integrands with \(x\)-bounds is invalid.
7. A denominator and its derivative suggest a logarithm
Check for this pattern before completing the square or decomposing a rational function. A numerator that is only partly related to \(g'(x)\) can often be rewritten as a derivative part plus a remainder.
8. Products may suggest integration by parts
This is especially useful when differentiating one factor makes it simpler, as with logarithmic, inverse-trigonometric, or polynomial factors multiplied by exponentials or trigonometric functions. Integration by parts is BC-specific content in this course sequence.
9. Choose \(u\) by the resulting integral
A useful choice makes \(du\) simpler and leaves \(dv\) easy to integrate. The LIATE mnemonic can suggest a starting point, but the real test is whether \(\int v\,du\) is easier than the original integral. Repeated polynomial factors may require repeated parts.
10. Classify rational functions by degree
For \(P(x)/Q(x)\), compare degrees before factoring. If \(\deg P\ge\deg Q\), perform polynomial long division. If \(\deg P<\deg Q\), the expression is proper and may be ready for substitution, completing the square, or partial fractions.
11. Long division can reveal several standard forms
Integrate the polynomial quotient directly, then classify the proper remainder. Skipping division often hides a much simpler solution.
12. Complete the square for quadratic denominators
A denominator with no convenient real factorization may be rewritten as \((x-h)^2+a^2\), revealing an inverse-tangent form:
First check whether the numerator contains the derivative of the quadratic; that part may instead produce a logarithm.
13. Use partial fractions for suitable proper rational functions
When a proper rational denominator factors into the forms covered by the course, decompose it into simpler rational terms. Distinct linear factors receive one constant-numerator term each. Partial fractions are BC-specific content here, and the decomposition should be verified algebraically.
14. Some integrals require a sequence of techniques
A substitution may create a rational integral; division may leave a term suited to partial fractions; rewriting a numerator may split one integral into logarithmic and inverse-tangent parts. State the purpose of each stage instead of forcing one method to do everything.
15. Improper setup surrounds the antiderivative method
An infinite bound or unbounded integrand determines the limit setup, not necessarily the inner antidifferentiation technique. First split at every improper point, then select an antiderivative method for each finite integral, and finally evaluate the limits.
16. Equivalent methods can both be valid
One student may simplify first while another uses substitution; their antiderivatives can differ by a constant and still be equivalent. Prefer the method with fewer transformations, but judge correctness by differentiation and domain, not by matching one exact form.
17. Differentiate to verify
Differentiate the complete result, including every coefficient and logarithmic absolute value. For a definite integral, also check sign and approximate size. Verification catches missing chain-rule factors and sign errors without repeating the entire solution.
18. AP selection routine
- Simplify and inspect the domain.
- Match a basic rule or reverse-chain pattern.
- If needed, classify products, rational degrees, and quadratic forms.
- Apply the selected method and any required follow-up method.
- Verify by differentiation and state the result in the requested form.
6. Detailed Worked Example and Error Check
Example 1: Simplify into basic rules.
No named transformation is needed because every term already matches a standard antiderivative.
Example 2: Recognize a reverse-chain pattern.
The numerator is exactly the derivative of the denominator, so substitution is more direct than any rational-function method.
Example 3: Divide before choosing the final rule.
Integrate the polynomial directly and use substitution on the remainder:
Example 4: Select integration by parts.
For \(\int xe^{2x}dx\), let \(u=x\) and \(dv=e^{2x}dx\). Then \(du=dx\) and \(v=e^{2x}/2\):
Example 5: Complete the square.
The numerator does not contain the derivative \(2x-6\), and the completed square reveals the inverse-tangent form.
Example 6: Factor and use partial fractions.
Therefore, on intervals avoiding the poles,
Example 7: Combine an improper limit with a basic method.
The infinite bound selects the limit framework; the finite integral itself uses a basic power rule after recognizing the shifted expression.
7. AP Reasoning Routine
Identify the accumulating quantity and units, preserve bounds, choose a valid integration technique, and check answers by differentiation.
- Identify the representation and requested quantity.
- State the rule or theorem and verify its conditions.
- Keep exact values until the final requested approximation.
- Interpret sign, units, interval, and context.
For each integral, name the first useful technique, justify the choice from its structure, and evaluate.
(a) \(\int(4x^3-3/x)\,dx\).
(b) \(\int x\cos(x^2)\,dx\).
(c) \(\int\ln x\,dx\), for \(x>0\).
(d) \(\int\frac{2x+5}{x^2+5x+7}\,dx\).
(e) \(\int\frac{dx}{x^2+4x+8}\).
(f) \(\int\frac{x^2+3}{x+1}\,dx\).
(g) \(\int\frac{2x+3}{(x-1)(x+2)}\,dx\).
(h) \(\int_0^1 3x^2e^{x^3}\,dx\).
(i) \(\int_1^\infty x^{-2}\,dx\).
(j) \(\int x^2e^x\,dx\).
Check the solution
(a) Use basic rules term by term: \(x^4-3\ln|x|+C\).
(b) Use substitution \(u=x^2\), since \(du=2x\,dx\): \(\frac12\sin(x^2)+C\).
(c) Treat the integrand as \((\ln x)(1)\) and use parts with \(u=\ln x\), \(dv=dx\): \(x\ln x-x+C\).
(d) The numerator is the derivative of the denominator, so substitution gives \(\ln(x^2+5x+7)+C\).
(e) Complete the square: \(x^2+4x+8=(x+2)^2+4\). The result is \(\frac12\arctan((x+2)/2)+C\).
(f) Long division gives \((x^2+3)/(x+1)=x-1+4/(x+1)\). The result is \(x^2/2-x+4\ln|x+1|+C\).
(g) Partial fractions give \(\frac{5/3}{x-1}+\frac{1/3}{x+2}\), so the result is \(\frac53\ln|x-1|+\frac13\ln|x+2|+C\).
(h) Substitute \(u=x^3\) and change the bounds from \(0,1\) to \(0,1\): \(\int_0^1e^u du=e-1\).
(i) The infinite bound requires a limit: \(\lim\limits_{b\to\infty}[-1/x]_1^b=1\), so the integral converges to \(1\).
(j) Apply integration by parts twice. The result is \(e^x(x^2-2x+2)+C\); differentiating confirms \(x^2e^x\).