What does this Calculus II: Integration Techniques cheat sheet cover?

It has 10 sections: Antiderivatives, u-Substitution, Parts, Trig Integrals, Trig Sub, Partial Fractions, Improper, Area/Volumes, Arc Length and Numerical. Together they cover the topics a typical Calculus II: Integration Techniques exam tests, including basic antiderivative table, u-substitution, integration by parts (LIATE) and trigonometric integrals.

Is the integration techniques cheat sheet free?

The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.

Can I bring this Calculus II: Integration Techniques cheat sheet to my exam?

Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.

The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 1 statement before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.

Can I make a Calculus II: Integration Techniques sheet from my own notes?

Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

MathCalculus II: Integration Techniques

Integration Techniques Cheat Sheet — 1-page PDF

Antiderivatives, substitution, parts, trig methods, partial fractions, applications, and numerical integration. The whole course on one dense, printable page for your midterm or final, with every formula typeset and worked examples where they help.

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For a specific exam: Calculus II Midterm study sheet.

1 pageUS Letter
10 sections11 topics
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Preview of the one-page Calculus II: Integration Techniques cheat sheet: Antiderivatives, substitution, parts, trig methods, partial fractions, applications, and numerical integration — the whole course, one page

On the sheet

What’s on it

The topics a typical Calculus II: Integration Techniques exam tests, all on the one page.

Basic antiderivative table
U-substitution
Integration by parts (LIATE)
Trigonometric integrals
Trig substitution
Partial fractions
Improper integrals
Area between curves
Volumes: disk, washer, shell
Arc length
Numerical integration (trapezoid, Simpson)

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The free part, as text

The free part of the sheet, section by section. Unlock the full sheet to read the rest, edit any block, and print it or save it as a PDF.

01Antiderivatives

Basic antiderivative table

$f (x)$	$\int f (x) d x$
$x^{n}$ $(n \neq = - 1)$	$\frac{x ^{n + 1}}{n + 1} + C$
$\frac{1}{x}$	$ln ∣ x ∣ + C$
$e^{x}$	$e^{x} + C$
$a^{x}$	$\frac{a ^{x}}{ln a} + C$ $(a > 0, a \neq = 1)$
$sin x$	$- cos x + C$
$cos x$	$sin x + C$
$sec^{2} x$	$tan x + C$
$csc^{2} x$	$- cot x + C$
$sec x tan x$	$sec x + C$
$csc x cot x$	$- csc x + C$

Antiderivative: $F^{'} (x) = f (x)$ on an interval; $\int f (x) d x = F (x) + C$ .

Linearity

\int (a f (x) + b g (x)) d x = a \int f (x) d x + b \int g (x) d x

$a, b$ constants

Substitution pattern

\int f (g (x)) g^{'} (x) d x = \int f (u) d u

set $u = g (x)$ , $d u = g^{'} (x) d x$

Watch out: $+ C$ only for indefinite integrals; definite integrals evaluate to numbers.

02Parts

Definite parts

\int_{a}^{b} u d v = [uv]_{a}^{b} - \int_{a}^{b} v d u

Choose $u$ by LIATE: Log, Inverse trig, Algebraic, Trig, Exponential.
Let $d v =$ remaining factor $d x$ .
Find $d u$ and $v = \int d v$ .
Apply formula; often repeats for $\int e^{x} sin x d x$ , $\int e^{x} cos x d x$ .

LIATE priority

Rank	Type
1	log
2	inverse trig
3	algebraic
4	trig
5	exponential

Watch out:Bad $u$ choice can make integral worse; avoid picking $u$ so $d u$ is more complicated than $u$ .

Ex $\int x e^{x} d x$

$u = x$ , $d v = e^{x} d x$ ; $d u = d x$ , $v = e^{x}$ ; $x e^{x} - \int e^{x} d x = x e^{x} - e^{x} + C$

03Trig Integrals

Key identities

sin^{2} x + cos^{2} x = 1, 1 + tan^{2} x = sec^{2} x, 1 + cot^{2} x = csc^{2} x

used to reduce powers

04Trig Sub

Substitute; transform $d x$ .
Simplify radical using trig identity.
Integrate in $θ$ .
Back-sub with triangle or identities.

Watch out:Need correct $θ$ -interval for inverse trig back-sub if asked for explicit antiderivative.

Ex $\int \frac{d x}{a ^{2} - x ^{2}}$

$x = a sin θ$ , $d x = a cos θ d θ$ , denom $= a cos θ$ ; integral $= \int d θ = θ + C = arcsin (x / a) + C$

05Partial Fractions

Proper rational function: $P (x) / Q (x)$ with $de g P < de g Q$ ; if improper, do long division first.

Linear factors

\frac{P ( x )}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}

distinct linear factors

Repeated factor

\frac{P ( x )}{( x - a ) ^{n}} = \frac{A _{1}}{x - a} + \dots + \frac{A _{n}}{( x - a ) ^{n}}

Irreducible quadratic

\frac{P ( x )}{x ^{2} + b x + c} = \frac{A x + B}{x ^{2} + b x + c}

and repeated powers analogously

06Area/Volumes

Crossing curves

A = \int_{a}^{c} (f - g) d x + \int_{c}^{b} (g - f) d x

split at intersections

Disk

V = π \int_{a}^{b} [R (x)]^{2} d x

slices perpendicular to axis

Washer

V = π \int_{a}^{b} (R (x)^{2} - r (x)^{2}) d x

hole radius $r$

Shell

V = 2 π \int_{a}^{b} (radius) (height) d x

slices parallel to axis

Method choice

axis	common slice
about $x$ -axis	$d x$ disks/washers
about $y$ -axis	$d x$ shells
about $y$ -axis	$d y$ disks/washers
about $x$ -axis	$d y$ shells

Watch out:For washer/disk, radii measured from axis of rotation; outer radius $R$ is farther from axis.

ExRegion between $y = x$ and $y = x^{2}$ on $[0, 1]$

$A = \int_{0}^{1} (x - x^{2}) d x = 1/6$

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Also on the full Calculus II: Integration Techniques sheet

10 more sections, with every definition, formula and worked example. Unlimited unlocks every full sheet in the library to edit, print or save as a PDF, and makes unlimited sheets from your own notes.

Antiderivatives

u-Substitution

Inner function
Definite u-sub
Common u-sub forms

Parts

Integration by parts

Trig Integrals

Power rules
Half-angle
Trig products

Trig Sub

Standard triangles
Substitution map

Partial Fractions

Improper

Improper integral
Infinite limit
Interior blow-up
Convergent
p-tests

Area/Volumes

Area between curves

Arc Length

Cartesian
Parametric
Polar

Numerical

Trapezoid
Simpson
Error signs
When to use

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FAQ

Questions

About the Calculus II: Integration Techniques sheet and making your own.

What does this Calculus II: Integration Techniques cheat sheet cover?: It has 10 sections: Antiderivatives, u-Substitution, Parts, Trig Integrals, Trig Sub, Partial Fractions, Improper, Area/Volumes, Arc Length and Numerical. Together they cover the topics a typical Calculus II: Integration Techniques exam tests, including basic antiderivative table, u-substitution, integration by parts (LIATE) and trigonometric integrals.
Is the integration techniques cheat sheet free?: The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.
Can I bring this Calculus II: Integration Techniques cheat sheet to my exam?: Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.
How accurate is it?: The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 1 statement before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.
Can I make a Calculus II: Integration Techniques sheet from my own notes?: Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

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