AP Calculus AB · Flashcards
Unit 7 — Integration Techniques & Review
Course framework: Unit 6 (integration techniques) and mixed review · integration by parts is BC only · 69 cards. Answer each card in your head, flip it, then mark it Got it or Review again. Your progress is saved in this browser.
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Integral Strategy✓ Known #1
What should you ask first when you see an integral?
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Answer#1
- Is it a basic power, exponential, log or trig antiderivative?
- Is it a chain-rule reversal (\(u\)-substitution)?
- If it is definite, can symmetry, properties or geometry finish it?
Trap: Starting random algebra before recognizing the structure.
Integral Strategy✓ Known #2
What is the classic signal for \(u\)-substitution?
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Answer#2
An inside function together with (a constant multiple of) its derivative elsewhere in the integrand.
Trap: Trying \(u\)-substitution on terms that are added rather than multiplied.
Integral Strategy✓ Known #3
The integrand is a rational function. What are the AB tools?
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Answer#3
- If the numerator’s degree is at least the denominator’s: long division first.
- An irreducible quadratic denominator: complete the square (often leads to \(\arctan\)).
- Partial fractions are BC only.
Trap: Forcing partial fractions on an AB problem.
Integral Strategy✓ Known #4
Evaluate \(\displaystyle\int \frac{x^2+x+1}{x}\,dx\).
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Answer#4
Divide first: \(x+1+\frac1x\). Antiderivative: \(\dfrac{x^2}{2}+x+\ln|x|+C\).
Trap: Trying \(u\)-substitution on the whole fraction.
Integral Strategy✓ Known #5
Evaluate \(\displaystyle\int \frac{1}{x^2+2x+2}\,dx\).
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Answer#5
Complete the square: \(x^2+2x+2=(x+1)^2+1\), so the integral is \(\arctan(x+1)+C\).
Trap: Using \(\ln|x^2+2x+2|\), which would need \(2x+2\) in the numerator.
Integral Strategy✓ Known #6
If you see \(\dfrac{f'(x)}{f(x)}\), what is the antiderivative?
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Answer#6
\(\ln|f(x)|+C\).
Trap: Dropping the absolute value.
Integral Strategy✓ Known #7
If you see \(f'(x)\,e^{f(x)}\), what is the antiderivative?
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Answer#7
\(e^{f(x)}+C\).
Trap: Multiplying by \(f(x)\) instead of recognizing the chain rule.
Integral Strategy✓ Known #8
If you see \(\dfrac{f'(x)}{\sqrt{f(x)}}\), what is the antiderivative?
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Answer#8
\(2\sqrt{f(x)}+C\), because \(\int u^{-1/2}\,du=2u^{1/2}+C\).
Trap: Integrating as if \(f(x)\) were \(x\).
Integral Strategy✓ Known #9
Before heavy algebra on a definite integral, what should you check?
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Answer#9
- Symmetry (even/odd function on \([-a,a]\)).
- Additivity or reversing the limits.
- Geometry for an exact area.
Trap: Expanding everything before checking properties.
Integral Strategy✓ Known #10
When an integral represents a rate in context, what must your answer include?
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Answer#10
An interpretation with units: rate × time = amount.
Trap: No units or no context sentence.
Integral Strategy✓ Known #11
What does “set up but do not evaluate” mean on an AP FRQ?
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Answer#11
Write a correct integral with its limits and integrand; you do not need to evaluate it.
Trap: Writing only “use \(u\)-substitution.”
Integral Strategy✓ Known #12
Definite integral with \(u=g(x)\): what is the clean practice?
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Answer#12
Either change the limits to \(u\)-limits, or substitute back to \(x\) before evaluating — and be consistent.
Trap: Changing the variable but keeping the old limits.
Integral Strategy✓ Known #13
What is the most common setup mistake in area and volume problems?
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Answer#13
Limits \(a\) and \(b\) that do not match the region.
Trap: Choosing 0 and 1 automatically.
Integral Strategy✓ Known #14
Your answer to an area or volume question is negative. What does that mean?
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Answer#14
The setup is wrong (top/bottom, outer/inner or limits); area and volume are never negative.
Trap: Leaving the negative answer.
Integral Strategy✓ Known #15
Your answer to a net-change problem is negative. Can it be correct?
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Answer#15
Yes — a negative net change is meaningful and should be interpreted.
Trap: Forcing an absolute value.
Integral Strategy✓ Known #16
Where does \(+C\) belong?
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Answer#16
Only on indefinite integrals. A definite integral is a number; the constant cancels.
Trap: Adding \(+C\) to a definite integral.
Integral Strategy✓ Known #17
Which evaluation line should you show after finding an antiderivative?
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Answer#17
\(\big[F(x)\big]_a^b=F(b)-F(a)\).
Trap: Writing \(F(a)-F(b)\).
Integral Strategy✓ Known #18
What does it mean to accumulate a rate from \(a\) to \(x\)?
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Answer#18
Compute \(\displaystyle\int_a^x \text{rate}(t)\,dt\): the net total from \(a\) up to \(x\).
Trap: Confusing the accumulated amount with the value of the rate.
u-Substitution Mastery✓ Known #19
\(u\)-substitution reverses which derivative rule?
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Answer#19
The chain rule.
Trap: Saying the product rule.
u-Substitution Mastery✓ Known #20
With \(u=g(x)\), what does \(\int f(g(x))\,g'(x)\,dx\) become?
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Answer#20
\(\int f(u)\,du\).
Trap: Leaving \(x\) in the integrand after substituting.
u-Substitution Mastery✓ Known #21
For \(\int 3x^2(1+x^3)^7\,dx\), choose \(u\) and find \(du\).
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Answer#21
- \(u=1+x^3\)
- \(du=3x^2\,dx\)
Trap: Choosing \(u=x^2\).
u-Substitution Mastery✓ Known #22
Evaluate \(\int 3x^2(1+x^3)^7\,dx\).
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Answer#22
With \(u=1+x^3\): \(\int u^7\,du=\dfrac{u^8}{8}+C=\dfrac{(1+x^3)^8}{8}+C\).
Trap: Forgetting to substitute back to \(x\).
u-Substitution Mastery✓ Known #23
Evaluate \(\displaystyle\int \frac{2x}{x^2+5}\,dx\).
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Answer#23
\(u=x^2+5\), \(du=2x\,dx\): \(\ln|x^2+5|+C\) \((=\ln(x^2+5)+C)\).
Trap: Writing \(\ln x\).
u-Substitution Mastery✓ Known #24
Why does \(\int \frac{1}{u}\,du\) give \(\ln|u|\)?
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Answer#24
\(\frac{d}{du}\ln|u|=\frac1u\) for both positive and negative \(u\), so the absolute value covers both cases.
Trap: Dropping the absolute value without justification.
u-Substitution Mastery✓ Known #25
Evaluate \(\int \cos(5x)\,dx\).
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Answer#25
\(u=5x\), \(du=5\,dx\): \(\dfrac{1}{5}\sin(5x)+C\).
Trap: Forgetting the \(\frac15\).
u-Substitution Mastery✓ Known #26
Evaluate \(\int e^{2x}\,dx\).
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Answer#26
\(\dfrac{1}{2}e^{2x}+C\).
Trap: Writing just \(e^{2x}\).
u-Substitution Mastery✓ Known #27
For \(\int (4x-1)\sqrt{2x^2-x+7}\,dx\), choose \(u\).
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Answer#27
\(u=2x^2-x+7\), since \(du=(4x-1)\,dx\). Result: \(\frac23(2x^2-x+7)^{3/2}+C\).
Trap: Choosing \(u=\sqrt{2x^2-x+7}\).
u-Substitution Mastery✓ Known #28
What is \(\int u^{1/2}\,du\)?
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Answer#28
\(\dfrac{2}{3}u^{3/2}+C\).
Trap: Multiplying by \(\frac32\) instead of dividing.
u-Substitution Mastery✓ Known #29
Evaluate \(\int_{0}^{1} 2x(x^2+1)^3\,dx\) by changing the limits.
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Answer#29
\(u=x^2+1\): \(u(0)=1,\ u(1)=2\). \(\displaystyle\int_1^2 u^3\,du=\left[\frac{u^4}{4}\right]_1^2=\frac{16-1}{4}=\frac{15}{4}\).
Trap: Keeping the limits 0 and 1 after switching to \(u\).
u-Substitution Mastery✓ Known #30
If you change to \(u\)-limits, do you substitute back to \(x\)?
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Answer#30
No — evaluate entirely in \(u\).
Trap: Mixing \(u\) and \(x\) at the evaluation step.
u-Substitution Mastery✓ Known #31
If you keep the \(x\)-limits, what must you do before evaluating?
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Answer#31
Substitute back to \(x\) so the limits match the variable.
Trap: Evaluating a \(u\)-expression at \(x\)-values.
u-Substitution Mastery✓ Known #32
After choosing \(u\), what must appear in the integrand?
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Answer#32
A factor matching \(du\) (up to a constant), so every \(x\) is replaced.
Trap: Leaving an extra \(x\) factor behind.
u-Substitution Mastery✓ Known #33
Antiderivative of \(\dfrac{f'(x)}{[f(x)]^2}\)?
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Answer#33
With \(u=f(x)\): \(\int u^{-2}\,du=-u^{-1}+C=-\dfrac{1}{f(x)}+C\).
Trap: Using \(\ln\) instead of the power rule.
u-Substitution Mastery✓ Known #34
Antiderivative of \(f'(x)\sin(f(x))\)?
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Answer#34
\(-\cos(f(x))+C\).
Trap: Missing the negative.
u-Substitution Mastery✓ Known #35
Antiderivative of \(f'(x)\cos(f(x))\)?
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Answer#35
\(\sin(f(x))+C\).
Trap: Swapping sine and cosine.
u-Substitution Mastery✓ Known #36
Antiderivative of \(f'(x)\sec^2(f(x))\)?
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Answer#36
\(\tan(f(x))+C\).
Trap: Writing \(\sec(f(x))\).
u-Substitution Mastery✓ Known #37
Antiderivative of \(\dfrac{f'(x)}{1+[f(x)]^2}\)?
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Answer#37
\(\arctan(f(x))+C\).
Trap: Using \(\ln\).
u-Substitution Mastery✓ Known #38
Antiderivative of \(\dfrac{f'(x)}{\sqrt{1-[f(x)]^2}}\)?
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Answer#38
\(\arcsin(f(x))+C\).
Trap: Using \(\arctan\).
u-Substitution Mastery✓ Known #39
Why is \(u\)-substitution so common in AB?
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Answer#39
Most AB integrals are built as chain-rule reversals rather than advanced algebraic decompositions.
Trap: Reaching for techniques the course does not require.
Trig Integrals✓ Known #40
\(\int \sin x\,dx\)?
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Answer#40
\(-\cos x + C\).
Trap: Forgetting the negative.
Trig Integrals✓ Known #41
\(\int \cos x\,dx\)?
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Answer#41
\(\sin x + C\).
Trap: Swapping sine and cosine.
Trig Integrals✓ Known #42
\(\int \sec^2 x\,dx\)?
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Answer#42
\(\tan x + C\).
Trap: Writing \(\sec x\).
Trig Integrals✓ Known #43
\(\int \csc^2 x\,dx\)?
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Answer#43
\(-\cot x + C\).
Trap: Missing the negative.
Trig Integrals✓ Known #44
\(\int \sec x\tan x\,dx\)?
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Answer#44
\(\sec x + C\).
Trap: Writing \(\tan x\).
Trig Integrals✓ Known #45
\(\int \csc x\cot x\,dx\)?
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Answer#45
\(-\csc x + C\).
Trap: A sign error.
Trig Integrals✓ Known #46
Power-reduction identity for \(\sin^2 x\)?
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Answer#46
\(\sin^2 x=\dfrac{1-\cos 2x}{2}\).
Trap: Using \(\frac{1+\cos 2x}{2}\).
Trig Integrals✓ Known #47
Power-reduction identity for \(\cos^2 x\)?
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Answer#47
\(\cos^2 x=\dfrac{1+\cos 2x}{2}\).
Trap: A sign error.
Trig Integrals✓ Known #48
Evaluate \(\int \sin^2 x\,dx\).
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Answer#48
Rewrite as \(\frac{1-\cos2x}{2}\): \(\dfrac{x}{2}-\dfrac{\sin 2x}{4}+C\).
Trap: Trying to integrate \(\sin^2x\) directly as \(\frac{\sin^3x}{3}\).
Trig Integrals✓ Known #49
Evaluate \(\int \sin(3x)\,dx\).
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Answer#49
\(-\dfrac{1}{3}\cos(3x)+C\).
Trap: Missing the \(\frac13\).
Trig Integrals✓ Known #50
Evaluate \(\int \sec^2(5x)\,dx\).
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Answer#50
\(\dfrac{1}{5}\tan(5x)+C\).
Trap: Forgetting the chain factor.
Trig Integrals✓ Known #51
Evaluate \(\displaystyle\int \frac{\cos x}{1+\sin x}\,dx\).
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Answer#51
\(u=1+\sin x\), \(du=\cos x\,dx\): \(\ln|1+\sin x|+C\).
Trap: Looking for a complicated trig substitution.
Properties / Symmetry / Geometry✓ Known #52
What is \(\int_a^b f + \int_b^c f\)?
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Answer#52
\(\int_a^c f\).
Trap: Forgetting orientation when the limits are reversed.
Properties / Symmetry / Geometry✓ Known #53
What is \(\int_b^a f\)?
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Answer#53
\(-\int_a^b f\).
Trap: Dropping the minus sign.
Properties / Symmetry / Geometry✓ Known #54
If \(\int_0^4 f(x)\,dx=7\), what is \(\int_4^0 f(x)\,dx\)?
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Answer#54
\(-7\).
Trap: Answering 7.
Properties / Symmetry / Geometry✓ Known #55
If \(f\) is even, what is \(\int_{-a}^a f(x)\,dx\)?
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Answer#55
\(2\int_0^a f(x)\,dx\).
Trap: Answering 0 (the odd-function rule).
Properties / Symmetry / Geometry✓ Known #56
If \(f\) is odd, what is \(\int_{-a}^a f(x)\,dx\)?
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Answer#56
\(0\).
Trap: Using the even-function rule.
Properties / Symmetry / Geometry✓ Known #57
The graph over an interval is a semicircle of radius \(r\) above the \(x\)-axis. What is the integral over its diameter?
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Answer#57
\(\dfrac{1}{2}\pi r^2\).
Trap: Using the circumference.
Properties / Symmetry / Geometry✓ Known #58
The region is a triangle with base \(b\) and height \(h\) above the \(x\)-axis. What is the integral?
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Answer#58
\(\dfrac{1}{2}bh\).
Trap: Forgetting the negative sign when the triangle is below the axis.
Properties / Symmetry / Geometry✓ Known #59
If \(f(x)\ge g(x)\) on \([a,b]\) with \(a<b\), which inequality holds?
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Answer#59
\(\displaystyle\int_a^b f(x)\,dx \ge \int_a^b g(x)\,dx\).
Trap: Reversing the inequality.
Properties / Symmetry / Geometry✓ Known #60
What is \(\int_a^a f(x)\,dx\)?
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Answer#60
\(0\).
Trap: Answering \(f(a)\).
Properties / Symmetry / Geometry✓ Known #61
State the linearity of integrals.
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Answer#61
\(\int (af+bg)=a\int f+b\int g\): integrals split over sums and constants factor out.
Trap: Thinking integrals split over products.
Integration by Parts (BC)✓ Known #62
BC only · Integration by parts formula?
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Answer#62
\(\int u\,dv = uv – \int v\,du\).
Trap: Forgetting the minus sign.
Integration by Parts (BC)✓ Known #63
BC only · Integration by parts reverses which derivative rule?
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Answer#63
The product rule.
Trap: Saying the chain rule.
Integration by Parts (BC)✓ Known #64
Is integration by parts on the AP Calculus AB exam?
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Answer#64
No. It is a BC-only topic; AB focuses on \(u\)-substitution and basic antiderivatives.
Trap: Spending AB study time on integration by parts.
Integration by Parts (BC)✓ Known #65
BC only · Which technique is used for \(\int x e^x\,dx\)?
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Answer#65
Integration by parts with \(u=x\), \(dv=e^x\,dx\): \(xe^x-e^x+C\).
Trap: Trying \(u\)-substitution, which does not work cleanly.
Common Traps✓ Known #66
When does \(\left|\int_a^b f\right|\) equal \(\int_a^b |f|\)?
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Answer#66
Only when \(f\) does not change sign on \([a,b]\).
Trap: Assuming they are always equal.
Common Traps✓ Known #67
Your setup uses \(dy\) but the limits are \(x\)-values. What is wrong?
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Answer#67
The limits must match the variable of integration: \(y\)-limits for \(dy\).
Trap: Forgetting to convert the limits.
Common Traps✓ Known #68
Total distance is asked and \(v\) changes sign. What must you do?
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Answer#68
Split at the zeros of \(v\) and integrate \(|v|\).
Trap: Taking \(\left|\int v\right|\) at the end.
Common Traps✓ Known #69
A definite integral comes out negative. What is the best habit?
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Answer#69
Interpret the sign in context (net change). If the question was about area or volume, revise the setup.
Trap: Applying an absolute value with no reasoning.
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