AP Calculus BC Formula Sheet: Every Formula You Need for Exam Day

Here's something that catches a lot of AP Calculus BC students off guard: the College Board doesn't give you a formula sheet on exam day. Unlike the AP Physics or AP Chemistry exams, there's no reference table. Every formula you need has to be in your head before you walk into that room on May 11th.

For BC, that's a lot of formulas — everything from AB plus integration techniques, series convergence tests, parametric equations, and polar coordinates.

Here's the complete list, organized by topic. Print this, tape it to your wall, and review it every time you study.

All the AB Formulas (You Still Need These)

BC includes everything from AB. If you need a refresher on basic derivative rules and integral formulas, I covered those in the essential calculus formulas post. The derivatives and integrals formula sheet has the printable version.

Quick summary of what you need from AB:

All derivative rules (power, product, quotient, chain)
Trig, exponential, logarithmic, and inverse trig derivatives
Basic antiderivatives and u-substitution
Fundamental Theorem of Calculus (both parts)
Riemann sums and the definition of the definite integral

Integration Techniques (BC-Specific)

Integration by Parts

Use LIATE to choose $u$ : Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Pick $u$ from whichever comes first.

Key applications:

$\int x e^x \, dx$ (let $u = x$ , $dv = e^x \, dx$ )
$\int \ln x \, dx$ (let $u = \ln x$ , $dv = dx$ )
$\int x^2 \sin x \, dx$ (by parts twice, or use tabular method)

Partial Fractions

For $\frac{P(x)}{Q(x)}$ where $\deg(P) < \deg(Q)$ :

Distinct linear factors: $\frac{A}{x-a} + \frac{B}{x-b}$
Repeated linear factors: $\frac{A}{x-a} + \frac{B}{(x-a)^2}$

If $\deg(P) \geq \deg(Q)$ , do polynomial long division first.

Improper Integrals

Infinite limits:

<div class="latex-block"> \int_a^{\infty} f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx </div>

Convergence benchmark: $\int_1^{\infty} \frac{1}{x^p} \, dx$ converges if $p > 1$ , diverges if $p \leq 1$ .

Series Convergence Tests

This is the section most students need to study hardest. Get the free sequences and series formula sheet for the condensed version.

Divergence Test: If $\lim_{n \to \infty} a_n \neq 0$ , the series diverges. (Can only prove divergence.)

Geometric Series: $\sum ar^n$ converges to $\frac{a}{1-r}$ if $|r| < 1$ .

p-Series: $\sum \frac{1}{n^p}$ converges if $p > 1$ .

Integral Test: If $f$ is positive, continuous, decreasing: $\sum f(n)$ and $\int f(x) \, dx$ converge/diverge together.

Comparison Tests:

Direct: $0 \leq a_n \leq b_n$ and $\sum b_n$ converges → $\sum a_n$ converges
Limit: $\lim \frac{a_n}{b_n} = L$ with $0 < L < \infty$ → both behave the same

Ratio Test: $L = \lim \left|\frac{a_{n+1}}{a_n}\right|$ — converges if $L < 1$ , diverges if $L > 1$ .

Root Test: $L = \lim \sqrt[n]{|a_n|}$ — same criteria as ratio test.

Alternating Series Test: $\sum (-1)^n b_n$ converges if $b_n$ is decreasing and $\lim b_n = 0$ .

Standard Maclaurin Series (Must Memorize)

<div class="latex-block"> e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots </div> <div class="latex-block"> \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots </div> <div class="latex-block"> \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots </div> <div class="latex-block"> \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots \quad (|x| < 1) </div> <div class="latex-block"> \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots </div>

Taylor Series (general form):

<div class="latex-block"> f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n </div>

Lagrange Error Bound:

where $M$ is the maximum of $|f^{(n+1)}|$ between $a$ and $x$ .

Parametric Equations

For a curve $x = f(t)$ , $y = g(t)$ :

First derivative: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

Second derivative: $\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$

Arc length: $L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$

Speed: $\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}$

Polar Coordinates

Area: $A = \frac{1}{2}\int_\alpha^\beta r^2 \, d\theta$

Arc length: $L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta$

Converting: $x = r\cos\theta$ , $y = r\sin\theta$ , $r^2 = x^2 + y^2$

Euler's Method

For $\frac{dy}{dx} = F(x, y)$ with step size $\Delta x$ :

$y_{n+1} = y_n + F(x_n, y_n) \cdot \Delta x$

Logistic Growth

<div class="latex-block"> \frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right) </div>

Solution: $P(t) = \frac{L}{1 + Ce^{-kt}}$

Maximum growth rate occurs at $P = \frac{L}{2}$ .

How to Study This

Staring at this list won't get it into your brain. Here's what will:

Write the formulas by hand every day for two weeks. No peeking. Check yourself after.
Practice with the FRQ Finder. Pull up BC-specific FRQs and work them under timed conditions. You'll quickly learn which formulas you reach for most.
Use the Calculus 2 course materials for additional practice on integration techniques and series.
Build your own formula card. A single index card with the formulas you struggle with most. Review it every night before bed.

The exam is May 11th. Start memorizing now, and by exam day these formulas will be second nature. If you want structured help, the Final Stretch program covers all of this with live instruction and practice exams.