Calculus 2 Formula Sheet: The Complete Reference Guide You Actually Need

Calc 2 throws more formulas at you than any other course in the calculus sequence. Integration techniques, convergence tests, parametric equations, polar coordinates — there's a lot to keep straight.

I put together this reference so you have everything in one place, organized the way it actually comes up in the course. For the printable, condensed version, grab the free Calculus 2 formula sheet. What follows here is the expanded walkthrough — the formulas plus context on when and how to use them.

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Integration Techniques

Integration by Parts

<div class="latex-block"> \int u \, dv = uv - \int v \, du </div>

Choose $u$ using LIATE: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Pick $u$ from whichever category comes first.

Common cases to recognize:

• $\int x e^x \, dx$ — let $u = x$, $dv = e^x \, dx$
• $\int \ln x \, dx$ — let $u = \ln x$, $dv = dx$
• $\int x^2 e^x \, dx$ — by parts twice (tabular method helps)

For $\int e^x \sin x \, dx$ and $\int e^x \cos x \, dx$: apply by parts twice, then solve for the original integral algebraically.

Trigonometric Integrals

Powers of sine and cosine:

• Odd power of $\sin x$: Save one $\sin x$, convert rest using $\sin^2 x = 1 - \cos^2 x$, sub $u = \cos x$
• Odd power of $\cos x$: Save one $\cos x$, convert using $\cos^2 x = 1 - \sin^2 x$, sub $u = \sin x$
• Both powers even: Use half-angle identities:

$\sin^2 x = \frac{1 - \cos 2x}{2}, \quad \cos^2 x = \frac{1 + \cos 2x}{2}$

Trigonometric Substitution

Expression	Substitution	Identity
$\sqrt{a^2 - x^2}$	$x = a \sin\theta$	$1 - \sin^2\theta = \cos^2\theta$
$\sqrt{a^2 + x^2}$	$x = a \tan\theta$	$1 + \tan^2\theta = \sec^2\theta$
$\sqrt{x^2 - a^2}$	$x = a \sec\theta$	$\sec^2\theta - 1 = \tan^2\theta$

Don't forget to convert back to $x$ at the end — draw the reference triangle.

Partial Fractions

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

• Distinct linear: $\frac{A}{x-a} + \frac{B}{x-b}$
• Repeated linear: $\frac{A}{x-a} + \frac{B}{(x-a)^2}$
• Irreducible quadratic: $\frac{Ax+B}{x^2+bx+c}$

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

Key Integral Formulas

<div class="latex-block"> \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan\frac{x}{a} + C </div> <div class="latex-block"> \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\frac{x}{a} + C </div>

$\int \sec x \, dx = \ln|\sec x + \tan x| + C$

$\int \csc x \, dx = -\ln|\csc x + \cot x| + C$

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>

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

<!-- IMAGE: A reference triangle diagram showing the three trig substitution cases -->

Sequences and Series

The sequences and series formula sheet has the condensed version. Here's the full breakdown.

Key Series

Geometric: $\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$ for $|r| < 1$

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

Convergence Tests Summary

Test	Use When	Conclusion
Divergence	Always check first	If $\lim a_n \neq 0$, diverges
Geometric	Series is $ar^n$	Converges if $\|r\| < 1$
p-Series	Series is $1/n^p$	Converges if $p > 1$
Integral	$f$ is positive, continuous, decreasing	Same as $\int f(x)dx$
Direct Comparison	Can bound $a_n$ by known series	Compare convergence/divergence
Limit Comparison	Looks like known series	$\lim a_n/b_n = L$ (finite, positive)
Ratio	Factorials, exponentials	$L < 1$: converges, $L > 1$: diverges
Root	$n$th powers	Same as ratio test criteria
Alternating Series	Alternating signs	$b_n$ decreasing and $\lim b_n = 0$

Standard Maclaurin Series

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

Taylor Series

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

Lagrange Error Bound: $|R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}$

Parametric Equations

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

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

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

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

Polar Coordinates

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

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

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

<!-- IMAGE: A polar coordinate graph with a curve and the area formula visually illustrated -->

How to Use This Reference

A formula sheet is only useful if you know when to apply each formula. My advice:

Practice with it, don't just read it. Have the sheet next to you while doing problems. Highlight the formulas you use most — those are the ones to memorize first.

Build it yourself first, then compare. Writing your own formula sheet is one of the best study activities. After you've made yours, check it against the printable Calc 2 formula sheet.

Organize by decision. Instead of "here are all the series tests," think "when I see a series, what's my first move?" That framework gets you through exams.

For additional practice, the Calculus 2 course materials cover each topic with notes and quizzes. And if the formulas make sense but you struggle applying them under pressure, that's exactly the kind of thing one-on-one tutoring can help with.