MATH 31B flash cards
46 cards for Integration and Infinite Series, laid out week by week and timed around your exams. Free on iPhone and the web.
Exam dates
Unblank times your reviews around these. You can always set your own dates in the app.
- FinalThu, Mar 18, 2027A sample date until the Registrar posts the real one
The first 20 cards
Read the question, think of your answer, then open the card.
Week 1 · Inverse functionsIf f(x) = x³ + x, what is (f⁻¹)′(2)?Show the answerHide the answer
1/4. Since f(1) = 2, (f⁻¹)′(2) = 1/f′(1) = 1/(3 + 1).
Week 1 · Logarithms and exponentialsWhat is the derivative of ln(x² + 1)?Show the answerHide the answer
2x/(x² + 1). The derivative of ln u is u′/u.
Week 1 · Logarithms and exponentialsWhy is ∫ 1/x dx written ln|x| + C rather than ln x + C?Show the answerHide the answer
1/x is defined for negative x too, and ln|x| has derivative 1/x on both sides of 0.
Week 1 · Logarithms and exponentialsWhat is the derivative of xˣ, for x > 0?Show the answerHide the answer
xˣ(ln x + 1). Take logs: ln y = x ln x, so y′/y = ln x + 1.
Neither the power rule nor the exponential rule applies, since both the base and the exponent vary.
Week 2 · Inverse trig functionsWhat is the derivative of arctan x?Show the answerHide the answer
1/(1 + x²).
Week 2 · Inverse trig functionsWhy does arcsin x only return angles from −π/2 to π/2?Show the answerHide the answer
sin x isn't one-to-one, so it is restricted to [−π/2, π/2], where it takes each value from −1 to 1 exactly once.
Week 2 · Inverse trig functionsWhat is ∫₀¹ 1/(1 + x²) dx?Show the answerHide the answer
π/4. An antiderivative is arctan x, and arctan 1 − arctan 0 = π/4.
Week 2 · Exponential growth and decayA quantity follows y′ = −0.2y. What is its half-life?Show the answerHide the answer
5 ln 2, about 3.47. Solve e^(−0.2t) = 1/2 to get t = ln 2 / 0.2.
Week 3 · Integration by partsWhat is the integration by parts formula?Show the answerHide the answer
∫ u dv = uv − ∫ v du.
It is the product rule for derivatives, integrated and rearranged.
Week 3 · Integration by partsWhat is ∫ x eˣ dx?Show the answerHide the answer
x eˣ − eˣ + C. Take u = x and dv = eˣ dx, so du = dx and v = eˣ.
Week 3 · Integration by partsWhat is ∫ ln x dx?Show the answerHide the answer
x ln x − x + C. Take u = ln x and dv = dx, so the new integral is just ∫ 1 dx.
Week 3 · Integration by partsWhat is ∫ eˣ sin x dx?Show the answerHide the answer
eˣ(sin x − cos x)/2 + C. Parts twice brings the original integral back, then solve for it.
Week 4 · Trigonometric integralsWhat is ∫ sin² x dx?Show the answerHide the answer
x/2 − sin(2x)/4 + C, using sin² x = (1 − cos 2x)/2.
Week 4 · Trigonometric integralsHow do you integrate sinᵐx cosⁿx when n is odd?Show the answerHide the answer
Save one cos x for du, turn the other cosines into sines with cos² x = 1 − sin² x, and let u = sin x.
Week 4 · Trigonometric substitutionWhich substitution fits an integral containing √(a² − x²)?Show the answerHide the answer
x = a sin θ, because a² − a² sin² θ = a² cos² θ, so the root becomes a cos θ.
With θ between −π/2 and π/2, cos θ ≥ 0, so no absolute value is needed.
Week 4 · Trigonometric substitutionWhat is ∫ 1/√(4 − x²) dx?Show the answerHide the answer
arcsin(x/2) + C. With x = 2 sin θ, the integrand reduces to ∫ dθ.
Week 4 · Trigonometric substitutionWhat is ∫₀² √(4 − x²) dx?Show the answerHide the answer
π. The graph is a quarter of a circle of radius 2, whose area is π(2)²/4.
Week 5 · Partial fractionsIn partial fractions, what comes first if the top's degree is at least the bottom's?Show the answerHide the answer
Polynomial long division, so the leftover fraction has a lower degree on top.
Week 5 · Partial fractionsWhat terms does a repeated factor (x − 2)² contribute to a partial fraction setup?Show the answerHide the answer
A/(x − 2) + B/(x − 2)², one term for each power.
Week 5 · Partial fractionsWhat term does an irreducible factor x² + 1 contribute to a partial fraction setup?Show the answerHide the answer
(Ax + B)/(x² + 1): a linear numerator over the quadratic.
Terms like this integrate to a logarithm plus an arctangent.
Week by week
The whole deck: 46 cards over 10 weeks. Every week is open from day one.
- 1Inverse functions and logarithms4 cards
- Inverse functions 1
- Logarithms and exponentials 3
- 2Inverse trig and exponential change4 cards
- Inverse trig functions 3
- Exponential growth and decay 1
- 3Integration by parts4 cards
- Integration by parts 4
- 4Trig integrals and trig substitution5 cards
- Trigonometric integrals 2
- Trigonometric substitution 3
- 5Partial fractions4 cards
- Partial fractions 4
- 6Improper integrals4 cards
- Improper integrals 4
- 7Sequences and series5 cards
- Sequences 2
- Geometric and telescoping series 3
- 8Convergence tests5 cards
- Divergence, p-series and comparison tests 3
- The ratio test 2
- 9Alternating and power series5 cards
- Alternating series 3
- Power series 2
- 10Taylor series6 cards
- Taylor and Maclaurin series 6