AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/6/6-14)
Unit 6 · Topic 6.14
6.14 Selecting Techniques for Antidifferentiation
Knowing many integration techniques only helps if you can tell which one fits. This topic is about reading an integrand and choosing a method (a basic rule, algebra, a trig identity, substitution, long division or completing the square, and in BC, parts or partial fractions) before you start computing.
Key terms
- antidifferentiation
- rewriting the integrand
- u-substitution
- long division
- integration techniques
A decision order that works
Run through these questions in order. Stop at the first yes.
- Is it a basic form, or a sum of basic forms after rewriting (roots as powers, splitting a fraction over a single term, expanding)? Use the basic rules (6.8).
- Is there an inner function whose derivative (up to a constant) is also a factor? Use u-substitution (6.9).
- Is it a fraction with top degree ≥ bottom degree? Divide first (6.10).
- Is the bottom an irreducible quadratic? Complete the square for arctan (6.10).
- Would a trig identity simplify it? For example, tan² x = sec² x − 1, or sin² x + cos² x = 1.
- BC: is it a product of two unrelated types, like x·eˣ or x·sin x, or a lone ln x? Use integration by parts (6.11).
- BC: is it a fraction whose bottom factors into different linear factors? Use partial fractions (6.12).
Look-alikes that need different methods
Small changes in the numerator change everything. With the denominator x² + 2x + 5:
| Integrand | Why | Method |
|---|---|---|
| (x + 1)/(x² + 2x + 5) | top is ½ of the bottom's derivative | u-substitution → ½ ln(x² + 2x + 5) |
| 1/(x² + 2x + 5) | no x on top; bottom doesn't factor | complete the square → arctan |
| x/(x² − 1) | top is ½ of the bottom's derivative | u-substitution (faster than partial fractions) |
| 1/(x² − 1) | bottom factors, top isn't its derivative | partial fractions (BC) |
| x e^(x²) | x is part of the derivative of x² | u-substitution |
| x eˣ | no inner function; product of types | integration by parts (BC) |
Tips for choosing quickly
- Look for a function and its derivative together: ln x with 1/x, sin x with cos x, x² with x, eˣ with eˣ, arctan x with 1/(1 + x²).
- If the integrand is a fraction, compare degrees before anything else.
- If you substitute and an x is left over, try solving for x in terms of u, or try another method.
- Many integrals can be done more than one way. Pick the fastest one you're confident in.
- If nothing works and a calculator is allowed, the question probably expects a numerical integral.
Check your answer
Differentiate your antiderivative. If you get the original integrand back, you're done. This takes seconds and catches most sign and constant errors.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Same denominator, different methods
Find (a) ∫ (x + 1)/(x² + 2x + 5) dx and (b) ∫ 1/(x² + 2x + 5) dx.
Show the solutionHide the solution
- Step 1: (a) The derivative of x² + 2x + 5 is 2x + 2 = 2(x + 1), and x + 1 is on top. Let u = x² + 2x + 5, du = 2(x + 1) dx.
- Step 2: Then the integral is ½ ∫ du/u = ½ ln|u| + C. Because x² + 2x + 5 = (x + 1)² + 4 is always positive, you can drop the absolute value: ½ ln(x² + 2x + 5) + C.
- Step 3: (b) No x on top, so substitution fails. Complete the square: (x + 1)² + 4 = (x + 1)² + 2².
- Step 4: ∫ 1/((x + 1)² + 2²) dx = ½ arctan((x + 1)/2) + C.
Answer: (a) ½ ln(x² + 2x + 5) + C (b) ½ arctan((x + 1)/2) + C
- Example 2
Trap: a trig identity, not a power rule
Find ∫ tan² x dx.
Show the solutionHide the solution
- Step 1: Tempting wrong answer: tan³ x/3 + C. That's not right, because the derivative of tan³ x/3 is tan² x · sec² x, not tan² x.
- Step 2: Use the identity tan² x = sec² x − 1.
- Step 3: ∫ (sec² x − 1) dx = tan x − x + C.
- Step 4: Check: d/dx (tan x − x) = sec² x − 1 = tan² x.
Answer: tan x − x + C
- Example 3BC only
BC: choosing parts
Find ∫ x ln x dx.
Show the solutionHide the solution
- Step 1: Substitution doesn't help: with u = ln x, du = (1/x) dx, but the integrand has a factor x, not 1/x. This is a product of an algebraic function and a log, so use parts.
- Step 2: LIATE: the log comes first, so u = ln x and dv = x dx. Then du = (1/x) dx and v = x²/2.
- Step 3: ∫ x ln x dx = (x²/2) ln x − ∫ (x²/2)(1/x) dx = (x²/2) ln x − ∫ (x/2) dx = (x²/2) ln x − x²/4 + C.
Answer: (x²/2) ln x − x²/4 + C
Common mistakes
- Using the power rule on a function raised to a power, like ∫ tan² x dx or ∫ (3x + 1)⁵ dx, without accounting for the inside.
- Reaching for logs whenever there's a fraction.
- BC: starting partial fractions when the top's degree isn't smaller than the bottom's. Divide first.
- Forgetting + C after all the work of choosing a method.
On the exam
- No-calculator multiple-choice questions often put several look-alike integrals side by side. Decide on the method from the structure before you compute anything.
- BC students should be ready for integrals that need parts or partial fractions, sometimes inside another question, such as a series or an improper integral.
Connected topics
Videos
Check yourself
4 questions on 6.14 Selecting Techniques for Antidifferentiation. Pick an answer to see if you got it, and why.
∫ (2x + 5)/(x² + 9) dx =
For which of the following integrals does the substitution u = x² + 1 turn the integral into a constant multiple of ∫ uⁿ du for some constant n?
∫ tan² x dx =
∫ e^(2x)/(eˣ + 1) dx =
0 of 4 answered