Integral Calculus Cheat Sheet

Integral Calculus Cheat Sheet By The WeSolveThem.com Team Simplicity is the ultimate sophistication. Leonardo da Vinci







Table of Contents Parametric and Polar Operations ....................................................................................................... 5 Notations ........................................................................................................................................................................................................ 5 First Derivative ............................................................................................................................................................................................ 5 Second Derivative ....................................................................................................................................................................................... 5 Trigonometric .............................................................................................................................................................................................. 5 Circle ................................................................................................................................................................................................................. 5 Ellipse ............................................................................................................................................................................................................... 5 Polar Derivative ........................................................................................................................................................................................... 6 Polar Equations for Ellipse ..................................................................................................................................................................... 6 Polar Equations for Hyperbola ............................................................................................................................................................. 6 Polar Equations for Parabola ................................................................................................................................................................ 6 Antiderivatives & Integration ............................................................................................................. 7 Basic Rules .................................................................................................................................................................................................... 7 Riemann Sum for Area Approximation ............................................................................................................................................ 7 Area Approximation Rules ..................................................................................................................................................................... 8 Midpoint Rule ................................................................................................................................................................................................ 8 Trapezoid Rule ............................................................................................................................................................................................. 8 The Integral Notation ∫ ..................................................................................................................... 8 Definite Integral Properties ................................................................................................................................................................... 8 Fundamental Theorems .......................................................................................................................................................................... 9 Limit Definition of a Definite Integral ............................................................................................................................................... 9 Differential Equation (1st order) .......................................................................................................................................................... 9 Common Integrals ................................................................................................................................................................................... 10 Definite Integral Rules .......................................................................................................................................................................... 10 Substitution ................................................................................................................................................................................................ 10 Integration by Parts ................................................................................................................................................................................ 10 Trig Substitution ...................................................................................................................................................................................... 11 Trig Identity ............................................................................................................................................................................................... 11 Partial Fractions ...................................................................................................................................................................................... 11 Integration Steps .............................................................................................................................. 12 Improper Integration ............................................................................................................................................................................ 12 Infinite Bounds .......................................................................................................................................................................................... 12 Undefined Bounds .................................................................................................................................................................................... 12 Areas, Volumes, and Curve Length ................................................................................................... 12 Area with respect to an axis ............................................................................................................................................................... 12 Cartesian ...................................................................................................................................................................................................... 12 Area between curves ............................................................................................................................................................................. 12 Polar Area ................................................................................................................................................................................................... 13 Volume about an axis (Disk Method) ............................................................................................................................................. 13 Volume between curves (Washer Method) ................................................................................................................................. 13 Cylindrical Shell Method ...................................................................................................................................................................... 13 Arc Length .................................................................................................................................................................................................. 13 Surface Area .............................................................................................................................................................................................. 14 Physics Applications .............................................................................................................................................................................. 14 Center of Mass with Constant Density ............................................................................................................................................. 14 Sequences vs Series .......................................................................................................................... 14

Sequence Tests ................................................................................................................................. 14 Series Tests ...................................................................................................................................... 15 Taylor series .............................................................................................................................................................................................. 16 Maclaurin Series ...................................................................................................................................................................................... 16 Power Series .............................................................................................................................................................................................. 16 Radius/Interval of Converges ............................................................................................................................................................ 16 3D Calculus ....................................................................................................................................... 17 Magnitude ................................................................................................................................................................................................... 17 Unit Vectors ............................................................................................................................................................................................... 17 Dot/Cross Product .................................................................................................................................................................................. 17 Dot .................................................................................................................................................................................................................. 17 Properties .................................................................................................................................................................................................... 17 Cross ............................................................................................................................................................................................................... 17 Properties .................................................................................................................................................................................................... 18 Angles Between Vectors ....................................................................................................................................................................... 18 Projections ................................................................................................................................................................................................. 18 Areas/Volume ........................................................................................................................................................................................... 18 Triangle ........................................................................................................................................................................................................ 18 Parallelogram ............................................................................................................................................................................................ 18 Parallelepiped ............................................................................................................................................................................................ 18 Line ................................................................................................................................................................................................................ 18 Line from tip to tip ................................................................................................................................................................................... 18 Equation of a Plane ................................................................................................................................................................................. 19 Vector Functions ..................................................................................................................................................................................... 19 Limit ............................................................................................................................................................................................................... 19 Derivative .................................................................................................................................................................................................... 19 Definite Integral ....................................................................................................................................................................................... 19 Indefinite Integral .................................................................................................................................................................................... 19 Differentiation Rules .............................................................................................................................................................................. 19 Arc length ................................................................................................................................................................................................... 19 Tangents ...................................................................................................................................................................................................... 20 Unit Tangent Vector ............................................................................................................................................................................... 20 Curvature 1 ................................................................................................................................................................................................. 20 Curvature 2 (vector function) ............................................................................................................................................................. 20 Curvature 3 (single variable) ............................................................................................................................................................ 20 Curvature 4 (parametric) ..................................................................................................................................................................... 20 Normal Vector ........................................................................................................................................................................................... 20 Binormal Vector ....................................................................................................................................................................................... 20 Tangential and Normal Components (acceleration) ............................................................................................................... 21 Physics Notations .................................................................................................................................................................................... 21 Position ......................................................................................................................................................................................................... 21 Velocity ......................................................................................................................................................................................................... 21 Speed .............................................................................................................................................................................................................. 21 Acceleration ................................................................................................................................................................................................ 21 Curvature ..................................................................................................................................................................................................... 21 Tangential Component (acceleration) ........................................................................................................................................... 21 Normal Component (acceleration) .................................................................................................................................................. 21 Acceleration ................................................................................................................................................................................................ 21 Note: .............................................................................................................................................................................................................. 21 Dot Product of Velocity and Acceleration ..................................................................................................................................... 21 Tangential Acceleration ........................................................................................................................................................................ 21

Normal Acceleration ............................................................................................................................................................................... 21 Frenet-Serret Formulas ......................................................................................................................................................................... 22

Partial Derivatives ............................................................................................................................ 22 Mixed Partial ............................................................................................................................................................................................. 22 Tangent Plane ........................................................................................................................................................................................... 22 Chain Rule .................................................................................................................................................................................................. 22 PreCalculus Review .......................................................................................................................... 23 Arithmetic ................................................................................................................................................................................................... 23 Exponential ................................................................................................................................................................................................ 23 Radicals ....................................................................................................................................................................................................... 23 Fractions ..................................................................................................................................................................................................... 23 Logarithmic ................................................................................................................................................................................................ 24 Other Formulas/Equations ................................................................................................................................................................. 24 Areas ............................................................................................................................................................................................................. 26 Surface Areas ............................................................................................................................................................................................ 26 Volumes ....................................................................................................................................................................................................... 26 Domain Restrictions .............................................................................................................................................................................. 26 Right Triangle ........................................................................................................................................................................................... 27 Reciprocal Identities ............................................................................................................................................................................... 27 Double Angle Formulas ........................................................................................................................................................................ 28 Half Angle Formulas ............................................................................................................................................................................... 28 Sum and Difference Formulas ............................................................................................................................................................ 28 Product to Sum Formulas ..................................................................................................................................................................... 28 Sum to Product Formulas ..................................................................................................................................................................... 28 Unit Circle ........................................................................................................................................ 29



Parametric and Polar Operations Notations 𝑥=𝑥 𝑡 , 𝑥! 𝑡 = First Derivative

Second Derivative

𝑡 ∈ 𝑎, 𝑏

𝑑𝑥 ≡ 𝑥 𝑑𝑡

𝑦=𝑦 𝑡 , 𝑦! 𝑡 =

𝑡 ∈ 𝑎, 𝑏

𝑑𝑦 ≡ 𝑦 𝑑𝑡

𝑑𝑦 𝑦! 𝑡 𝑑𝑡 = 𝑑𝑦 ⋅ 𝑑𝑡 = 𝑑𝑦 = 𝑑𝑥 𝑥! 𝑡 𝑑𝑡 𝑑𝑥 𝑑𝑥 𝑑𝑡

𝑑 𝑑 𝑑 𝑑𝑦 𝑑 𝑑𝑥 𝑥 ! 𝑡 𝑑𝑥 𝑦 ! 𝑡 − 𝑦 ! 𝑡 𝑑𝑥 𝑥 ! 𝑡 𝑥 ! 𝑡 𝑑𝑥 𝑑𝑡 – 𝑦 ! 𝑡 𝑑𝑥 𝑑𝑡 𝑑! 𝑦 𝑑 𝑑𝑦 𝑑 𝑦! 𝑡 = = = = 𝑑𝑥 ! 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑥 ! 𝑡 𝑥! 𝑡 ! 𝑥! 𝑡 ! 𝑑 𝑑𝑦 𝑑 𝑑𝑥 𝑑 𝑑𝑦 𝑑 𝑑 𝑑𝑦 𝑑 𝑑𝑦 𝑥 ! 𝑡 𝑑𝑡 𝑑𝑥 − 𝑦 ! 𝑡 𝑑𝑡 𝑑𝑥 𝑥 ! 𝑡 𝑑𝑡 𝑑𝑥 − 𝑦 ! 𝑡 𝑑𝑡 1 𝑥 ! 𝑡 𝑑𝑡 𝑑𝑥 𝑑𝑡 = = = = ! 𝑑𝑥 ! ! ! ! ! ! 𝑥 𝑡 𝑥 𝑡 𝑥 𝑡 𝑥 𝑡 𝑑 𝑑𝑦 𝑑 ! 𝑦 𝑑𝑡 𝑑𝑥 ∴ != 𝑑𝑥 𝑑𝑥 𝑑𝑡 Trigonometric 𝑦 𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 sin 𝜃 𝑥! + 𝑦! = 𝑟! 𝜃 = arctan 𝑥 Circle 𝑥−ℎ ! 𝑦−𝑘 ! 𝑥−ℎ 𝑦−𝑘 + = 1 = cos 𝜃 ! + sin 𝜃 ! ⇒ = cos 𝜃 ∧ = sin 𝜃 , 𝜃 ∈ 0, 2𝜋 𝑟 𝑟 𝑟 𝑟 Ellipse 𝑥−ℎ ! 𝑦−𝑘 ! 𝑥−ℎ 𝑦−𝑘 + = 1 = cos 𝜃 ! + sin 𝜃 ! ⇒ = cos 𝜃 ∧ = sin 𝜃 , 𝜃 ∈ 0, 2𝜋 𝑎 𝑏 𝑎 𝑏

Polar Derivative

𝑑𝑦 𝑑 𝑑𝑦 𝑑𝜃 𝑑𝜃 𝑟 sin 𝜃 𝑟 𝜃 cos 𝜃 + 𝑟 ! 𝜃 sin 𝜃 = = = ! 𝑑 𝑑𝑥 𝑑𝑟 𝑟 𝜃 cos 𝜃 − 𝑟 𝜃 sin 𝜃 𝑑𝜃 𝑑𝜃 𝑟 cos 𝜃

Polar Equations for Ellipse 𝑥! 𝑦! + = 1 𝑎! 𝑏! 𝑥! 𝑦! + = 1 𝑏! 𝑎! 𝑒 < 1

𝑒𝑑 𝑟 𝜃 = 𝑎 ± 𝑒 cos 𝜃

0 ≤ 𝑎 < 𝑏

𝑒 = eccentricity, 𝑑 = diretrix 𝑒𝑑 𝑟 𝜃 = 𝑎 ± 𝑒 sin 𝜃

Polar Equations for Hyperbola 𝑥! 𝑦! − = 1 𝑎! 𝑏!

𝑒𝑑 𝑟 𝜃 = 𝑎 ± 𝑒 cos 𝜃 Polar Equations for Parabola 𝑒 = 1 𝑟 𝜃 =

𝑑 𝑎 ± cos 𝜃

𝑐 ! = 𝑎! − 𝑏! 𝑐 𝑒= 𝑎

𝑐 ! = 𝑎! + 𝑏! Foci ±𝑐, 0 Vertices ±𝑎, 0 ! Asymptotes 𝑦 = ± ! 𝑥 𝑐 ! = 𝑎! + 𝑏! Foci 0, ±𝑐, Vertices 0, ±𝑎, ! Asymptotes 𝑦 = ± ! 𝑥

𝑦! 𝑥! − = 1 𝑎! 𝑏!

𝑒 > 11

𝑐 ! = 𝑎! − 𝑏! Foci ±𝑐, 0 Vertices ±𝑎, 0 𝑐 ! = 𝑎! − 𝑏! Foci 0, ±𝑐, Vertices 0, ±𝑎,

0 ≤ 𝑎 < 𝑏

𝑒 = eccentricity, 𝑑 = diretrix 𝑒𝑑 𝑟 𝜃 = 𝑎 ± 𝑒 sin 𝜃

𝑒 = eccentricity, 𝑑 = diretrix 𝑟 𝜃 =

𝑑 𝑎 ± sin 𝜃

𝑐 ! = 𝑎! + 𝑏! 𝑐 𝑒= 𝑎

𝑦 ! = 4𝑝𝑥, 𝑥 ! = 4𝑝𝑦,

𝑑 = −𝑝 𝑑 = −𝑝



Antiderivatives & Integration Basic Rules Power Rule for antiderivatives

1 𝑥 ! + 𝐶 ⇔ 𝑛 ≠ −1 𝑛+1 𝑎! ⇒ 𝑦= + 𝐶 ln 𝑎

𝑦! = 𝑥! ⇒ 𝑦 =

Exponential

𝑦! = 𝑎!

1 ⇒ 𝑦 = ln 𝑥 + 𝐶 𝑥 1 1 𝑦! = ⇒ 𝑦 = ln 𝑎𝑥 + 𝑏 + 𝐶 𝑎𝑥 + 𝑏 𝑎

Natural Log (case 1)

𝑦! =

Natural Log (case 2)

𝑢! 𝑥 𝑦 = ⇒ 𝑦 = ln 𝑢 𝑥 + 𝐶 𝑢 𝑥

Natural Log (case 3)

!

Euler’s Number (case 1)

1 !" 𝑒 + 𝐶 𝑎 1 ⇒ 𝑦 = 𝑒 !"!! + 𝐶 𝑎

𝑦 ! = 𝑒 !" ⇒ 𝑦 =

Euler’s Number (case 2)

𝑦 ! = 𝑒 !"!!

Euler’s Number (case 3)

𝑦 ! = 𝑢! 𝑥 𝑒 !

Anti-Chain-Rule Substitution Method

𝑦 ! = 𝑓 ! 𝑔 𝑥 𝑔! 𝑥 ⇒ 𝑦 = 𝑓 𝑔 𝑥



!

⇒ 𝑦 = 𝑒!

!

+ 𝐶 + 𝐶



Riemann Sum for Area Approximation !

𝑓 𝑥!∗ 𝛥𝑥 ,

𝐴 ≈ lim

!→∞



!!!

!

𝛥𝑥 =

𝑏−𝑎 , 𝑛

𝑥! = 𝑎 + 𝑖 ∙ 𝛥𝑥 !

𝑐 = 𝑐𝑛

𝑖=

!!!

!!!

!

!

𝑐𝑓 𝑥! = 𝑐 !!!

!

!!!

!!!

!

!!!



𝑖! =

𝑓 𝑥! !

𝑓 𝑥! ± 𝑔 𝑥!

𝑛 𝑛+1 2

=

!

𝑓 𝑥! ± !!!

!

!!!

𝑛 𝑛+1 𝑖 = 2 !

𝑔 𝑥! !!!

𝑛 𝑛 + 1 2𝑛 + 1 6 !



Area Approximation Rules Midpoint Rule ! !

Trapezoid Rule

𝑏−𝑎 𝑥! + 𝑥! 𝑥! + 𝑥! 𝑓 𝑥 𝑑𝑥 ≈ 𝑓 +𝑓 +⋯ 𝑛 2 2

!

𝑓 𝑥 𝑑𝑥 ≈ !



𝑏−𝑎 𝑓 𝑥! + 2𝑓 𝑥! + 2𝑓 𝑥! + ⋯ + 2𝑓 𝑥!!! + 𝑓 𝑥! 2𝑛

The Integral Notation ∫ !

𝑓(𝑥!∗ ) 𝛥𝑥 ≡

lim

!→∞



!!!

Definite Integral Properties

!

𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎

𝑐 𝑑𝑥 = 𝑐 𝑏 − 𝑎

!

!

!

!

𝑓 𝑥 𝑑𝑥 = 0

!

𝑐𝑓 𝑥 𝑑𝑥 = 𝑐

!

!

!

𝑓 𝑥 𝑑𝑥 !

!

𝑓 𝑥 𝑑𝑥 = 0

!

𝑓 𝑥 ± 𝑔 𝑥 𝑑𝑥 =

!!

!

⇔ 𝑓 −𝑥 = −𝑓 𝑥 ! !!

𝒌

𝑓 𝑥 𝑑𝑥 =

!

!

𝑔 𝑥 𝑑𝑥 !

!

𝑓 𝑥 𝑑𝑥 + !

𝑔 𝑥 𝑑𝑥 ⋅

𝑓 𝑥 𝑑𝑥

!

!

𝑓 𝑥 𝑑𝑥 = − !

𝑓 𝑥 𝑑𝑥 𝒌

even

NOTE: 𝑓 𝑥 ⋅ 𝑔 𝑥 𝑑𝑥 ≠

!

!

𝑓 𝑥

⇔ 𝑓 −𝑥 = 𝑓 𝑥

!

𝑓 𝑥 𝑑𝑥 ±

odd

!

𝑓 𝑥 𝑑𝑥 = 2



𝑓(𝑥) 𝑑𝑥 !



!



!

𝑓 𝑥 𝑑𝑥 !

Fundamental Theorems



Let 𝑓 𝑥 = 𝑢 and 𝑔 𝑥 = 𝑣 for the following: !

𝑖)

𝑦=

𝑓 𝑡 𝑑𝑡 ⇒

𝑦 ! = 𝑓 𝑣 ∙ 𝑣 ! − 𝑓 𝑢 ∙ 𝑢′

!



!

𝑦=

𝑦 ! = 𝑓 𝑣 ∙ 𝑣 ! − 𝑓 𝑎 ∙ 𝑎! = 𝑓 𝑣 ∙ 𝑣 ! − 0 = 𝑓 𝑣 ∙ 𝑣 !

𝑓 𝑡 𝑑𝑡 ⇒ !



!

𝑦=

𝑦 ! = 𝑓 𝑏 ∙ 𝑏 ! − 𝑓 𝑢 ∙ 𝑢! = 0 − 𝑓 𝑢 ∙ 𝑢! = −𝑓 𝑢 ∙ 𝑢′

𝑓 𝑡 𝑑𝑡 ⇒ !



Limit Definition of a Definite Integral !

𝑖𝑖)

𝑓(𝑥!∗ ) 𝛥𝑥

lim

!→∞

!!!

𝛥𝑥 =

𝑏−𝑎 , 𝑛

!

=

𝑓(𝑥) 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎 !

𝑥! = 𝑎 + 𝑖 ∙ 𝛥𝑥

Differential Equation (1st order)

𝑑𝑦 = 𝑓 ! 𝑥 ⇒ 𝑑𝑦 = 𝑓 ! 𝑥 𝑑𝑥 ⇒ 𝑑𝑦 = 𝑓 ! 𝑥 𝑑𝑥 𝑑𝑥 ⇒ 𝑦 + 𝑐! = 𝑓 𝑥 + 𝑐! ⇒ 𝑦 = 𝑓 𝑥 + 𝑐! − 𝑐! = 𝑓 𝑥 + 𝑐! ≡ 𝑓 𝑥 + 𝐶 𝑦! = 𝑓! 𝑥 ⇒



Common Integrals 𝑑𝑥 = 𝑥 + 𝐶

𝑘 𝑑𝑥 = 𝑘𝑥 + 𝐶

1 𝑥 ! 𝑑𝑥 = 𝑥 ! + 𝐶 3

𝑥 ! 𝑑𝑥 =

1 𝑥 𝑑𝑥 = 𝑥 ! + 𝐶 2

1 𝑥 !!! + 𝐶 𝑛+1

1 𝑑𝑥 = ln |𝑥| + 𝐶 𝑥

⇔ 𝑛 ≠ −1 1 !" 𝑒 + 𝐶 𝑎

𝑒 ! 𝑑𝑥 = 𝑒 ! + 𝐶

𝑒 !" 𝑑𝑥 =

1 𝑑𝑥 = ln 𝑥 + 1 + 𝐶 𝑥+1

1 1 𝑑𝑥 = ln 𝑎𝑥 + 𝑏 + 𝐶 𝑎𝑥 + 𝑏 𝑎

𝑒 ! 𝑢′ 𝑑𝑢 = 𝑒 ! + 𝐶

𝑢! 𝑑𝑢 = ln 𝑢 + 𝐶 𝑢

𝑒 !"!! 𝑑𝑥 =

1 !"!! 𝑒 + 𝐶 𝑎

𝑓 𝑢 𝑢′ 𝑑𝑢 = 𝐹 𝑢 + 𝐶 !

𝑓 𝑥 =𝐹 𝑏 −𝐹 𝑎 !

𝑢! cos 𝑢 𝑑𝑢 = sin 𝑢 + 𝐶

𝑢! sin 𝑢 𝑑𝑢 = − cos 𝑢 + 𝐶

𝑢! sec ! 𝑢 𝑑𝑢 = tan 𝑢 + 𝐶

𝑢! csc 𝑢 sec 𝑢 𝑑𝑢 = − csc 𝑢 + 𝐶

𝑢! sec 𝑢 tan 𝑢 𝑑𝑢 = sec 𝑢 + 𝐶

𝑢! csc ! 𝑢 𝑑𝑢 = − cot 𝑢 + 𝐶

𝑢! 1 − 𝑢!

𝑑𝑢 = arcsin 𝑢 + 𝐶

−𝑢! 1 − 𝑢!

𝑢! 𝑑𝑢 = arctan 𝑢 + 𝐶 1 + 𝑢!

𝑑𝑢 = arccos 𝑢 + 𝐶

Definite Integral Rules Substitution

!

Integration by Parts

𝑓 𝑔 𝑥 𝑔! 𝑥 𝑑𝑥 =

! !

! !

𝑓 𝑢 𝑑𝑢 ! !

𝑓 𝑥 𝑔! 𝑥 𝑑𝑥 = 𝑓 𝑥 𝑔 𝑥

!

Let 𝑢=𝑓 𝑥 𝑑𝑢 = 𝑓 ! 𝑥 𝑑𝑥 Then !

𝑢 𝑑𝑣 = 𝑢𝑣



!

! !

! !

!

−

𝑔 𝑥 𝑓 ! 𝑥 𝑑𝑥

!

𝑑𝑣 = 𝑔! 𝑥 𝑑𝑥 𝑣=𝑔 𝑥

!

−

𝑣 𝑑𝑢 !

Trig Substitution 𝑎! − 𝑥 !

𝑎! + 𝑥 !

𝑥 ! − 𝑎!

1 − sin! 𝜃 = cos ! 𝜃

1 + tan! 𝜃 = sec ! 𝜃

sec ! 𝜃 − 1 = tan! 𝜃

𝑥 = 𝑎 sin 𝜃 𝜋 𝜋 𝜃∈ − , 2 2

𝑥 = 𝑎 tan 𝜃 𝜋 𝜋 𝜃∈ − , 2 2

𝑥 = 𝑎 sec 𝜃

Trig Identity tan 𝑥 𝑑𝑥 =

sin 𝑥 𝑑𝑥 = − cos 𝑥

𝜃 ∈ 0,



1 ⋅ − sin 𝑥 𝑑𝑥, cos 𝑥

𝑑 ln 𝑢 𝑥 𝑑𝑥

= − ln cos 𝑥 + 𝐶 = ln Partial Fractions 𝑝 𝑥 𝐴 𝐵 = + 𝑥 𝑥+1 𝑥 𝑥+1 𝑝 𝑥 𝐴 𝐵𝑥 + 𝐶 = + ! ! 𝑥 𝑥 +1 𝑥 𝑥 +1



=

𝜋 3𝜋 ∨ 𝜃 ∈ 𝜋, 2 2

1 𝑑𝑢 𝑢 𝑑𝑥

1 + 𝐶 = ln sec 𝑥 + 𝐶 cos 𝑥

𝑝 𝑥 𝑥! 𝑥 + 1 𝑝 𝑥 𝑥 𝑥! + 1

!

=

𝐴 𝐵 𝐶 + !+ 𝑥 𝑥 𝑥+1

=

𝐴 𝐵𝑥 + 𝐶 𝐷𝑥 + 𝐸 + ! + ! 𝑥 𝑥 +1 𝑥 +1 !



Integration Steps

Ask yourself the following questions: 1. Is the integrand in integratable form? 2. Can I perform a function or trig-identity manipulation? 3. Should I use U-Substitution or Trig-Substitution? 4. Integration by Parts? 5. Partial fraction decomposition?



For a definite integral always check to see if the function is defined on the bounds Improper Integration Infinite Bounds !∞

!

𝑓 𝑥 𝑑𝑥 = !∞

Undefined Bounds

!∞

𝑓 𝑥 𝑑𝑥 + !∞

!

𝑓 𝑥 𝑑𝑥 = lim

!! →!∞ ! !

!

! !

!

𝑓 𝑥 𝑑𝑥, 𝑥 ∈ 𝑎, 𝑏 ⇒ lim! !! →𝑎

!!

𝑓 𝑥 𝑑𝑥 + lim

𝑓 𝑥 𝑑𝑥 + lim! !! →!



!!

!! → ∞ !

!!

𝑓 𝑥 𝑑𝑥

𝑓 𝑥 𝑑𝑥

!

Areas, Volumes, and Curve Length Area with respect to an axis Cartesian 𝑥 − 𝑎𝑥𝑖𝑠

𝑦 − 𝑎𝑥𝑖𝑠

!

𝐴= !

!

𝑓 𝑥 𝑑𝑥 ⇔ 𝑓 𝑥 ≥ 0 ∀! ∈ 𝑎, 𝑏

𝐴= !

𝑔 𝑦 𝑑𝑦 ⇔ 𝑔 𝑦 ≥ 0 ∀! ∈ 𝑐, 𝑑

Area between curves Given two curves 𝑓 ∧ 𝑔 set them equal to each other to find all x-coordinates of intersection. !

𝐴= !

𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥 ⇔ 𝑓 𝑥 ≥ 𝑔 𝑥 ∀! ∈ 𝑎, 𝑏

! !!!

𝐴=

𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥 = !!



or !! !!

𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥 +

!! !!

𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥 + ⋯

Polar Area 1 𝐴= 2

!!

!

𝑟 𝜃

!!

1 𝑑𝜃 ∧ 𝐴 = 2

!!

𝑅 𝜃

!

!

− 𝑟 𝜃

𝑑𝜃

!!



Volume about an axis (Disk Method) 𝑥 − 𝑎𝑥𝑖𝑠 !

𝑉=𝜋

𝑓 𝑥 !

!

𝑦 − 𝑎𝑥𝑖𝑠 !

𝑑𝑥 ⇔ 𝑓 𝑥 ≥ 0 ∀! ∈ 𝑎, 𝑏

𝑉=𝜋

𝑔 𝑦 !

!

𝑑𝑦 ⇔ 𝑔 𝑦 ≥ 0 ∀! ∈ 𝑐, 𝑑

Volume between curves (Washer Method) Given two curves 𝑓 ∧ 𝑔 set them equal to each other to find all x-coordinates of intersection. !

𝑉=𝜋 Cylindrical Shell Method Rotate about 𝑦 − 𝑎𝑥𝑖𝑠

!

𝑓 𝑥

− 𝑔 𝑥

!

!

Rotate about 𝑥 − 𝑎𝑥𝑖𝑠

!

𝑉=

!

2𝜋𝑥𝑓 𝑥 𝑑𝑥

𝑉=

!



Arc Length Cartesian 1 − 𝑓! 𝑥 !



2𝜋𝑦𝑔 𝑦 𝑑𝑦 !

Polar

!

𝐿=

𝑑𝑥 ⇔ 𝑓 𝑥 ≥ 𝑔 𝑥 ∀! ∈ 𝑎, 𝑏

!

𝑑𝑥

𝐿=

!!

!!

Parametric 𝑟 𝜃

!

− 𝑟! 𝜃

!

𝑑𝜃

𝐿=

!!

!!

𝑥! 𝑡

!

− 𝑦! 𝑡

!

𝑑𝑡

Surface Area Cartesian

Polar

𝑆!!!"#$ =

𝑆!!!"#$ =

!

𝑑𝑙 =

2𝜋𝑓 𝑥 𝑑𝑙, !

1−

𝑥

! 𝑑𝑥

!

𝑆!!!"#$ = 𝑑𝑙 =

𝑓!

2𝜋𝑔 𝑦 𝑑𝑙, !

𝑔!

1−

𝑦

! 𝑑𝑦

𝑑𝑙 =

!

𝑀! = 𝜌

− 𝑟! 𝜃

!

𝑑𝑙 =

! 𝑑𝜃

2𝜋𝑟 𝜃 sin 𝜃 𝑑𝑙

!

− 𝑟! 𝜃

𝑑𝑙 =

! 𝑑𝜃

!

𝑥=

1 𝐴

𝑓 𝑥 𝑑𝑥, 𝑓 𝑥 ≥ 0 ∈ 𝑎, 𝑏

!!

− 𝑦! 𝑡

!

2𝜋𝑥 𝑡 𝑑𝑙,

!!

𝑥! 𝑡

!

− 𝑦! 𝑡

! 𝑑𝜃

𝑑𝑥

!

𝑚 = 𝜌𝐴 = 𝜌

!

1 ∴𝑦= 2𝐴

𝑥 𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥, 𝑓 ≥ 𝑔 ∈ 𝑎, 𝑏

𝑀! =

𝑥𝑓 𝑥 𝑑𝑥 !

!

𝑓 𝑥 𝑑𝑥, 𝑓 𝑥 ≥ 0 ∈ 𝑎, 𝑏

1 2𝐴

!

𝑓 𝑥

!

𝑑𝑥

𝑓 𝑥

!

− 𝑔 𝑥

! !

!

𝑑𝑥, 𝑓 ≥ 𝑔 ∈ 𝑎, 𝑏

!

Sequences vs Series Series

∞

𝑎! = 𝑎! + 𝑎! + 𝑎! + ⋯

𝑎! = 𝑎! , 𝑎! , 𝑎! , …

𝑎! Converges

! 𝑑𝜃

!

Sequence

!!!

Sequence Tests lim 𝑎! = 𝐿

!→∞



!

!





!

𝑓 𝑥

!

1 ∴𝑥= 𝐴

2𝜋𝑦 𝑡 𝑑𝑙,

!!

𝒚-coordinate 𝑀! 𝑦= 𝑚

!

𝑚 = 𝜌𝐴 = 𝜌

!!

𝑥! 𝑡

𝑆!!!"#$ =

!!

𝜌 𝑀! = 2

𝑥𝑓 𝑥 𝑑𝑥

𝑆!!!"#$ =

2𝜋𝑟 𝜃 cos 𝜃 𝑑𝑙

!

!!

𝑟 𝜃

Physics Applications Center of Mass with Constant Density 𝒙-coordinate 𝑀! 𝑥= 𝑚

Parametric

!!

𝑟 𝜃

𝑆!!!"#$ = 𝑑𝑙 =

!!

𝑎! Diverges

lim 𝑎! = ±∞ ∨ 𝐷𝑁𝐸

!→∞



Series Tests Test Geometric P-Series Integral Test

∞

Form 𝑎𝑟 !!!

!!! ∞

!!! ∞

1 𝑛! 𝑎!

Condition

Diverges 𝒓 ≥ 𝟏



𝒑 ≤ 𝟏

∞

𝒇 𝒙 𝒅𝒙 = ∞

𝟏

𝒆. 𝒈. 𝒂𝒏 = ⇒𝒇 𝒙 = Comparison

𝑎!

𝒂𝒏 , 𝒃𝒏 𝐚𝐫𝐞 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞

!!!

Limit Comparison Alternating Series Ratio Root

𝒂𝒏 , 𝒃𝒏 𝐚𝐫𝐞 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 𝑎! 𝒂𝒏 !!! 𝐥𝐢𝐦 = 𝒌, 𝒌 > 𝟎 𝒏→∞ 𝒃𝒏 ∞ −1 !!! 𝑐! 𝒄𝒏 > 𝟎 !!! ∞

If

𝟏 𝒙𝟐

𝒂𝒏 ≥ 𝒃𝒏 ∀𝒏 ⇔ 𝒃𝒏 𝐝𝐢𝐯𝐞𝐫𝐠𝐞𝐬

𝒂𝒏 ≤ 𝒃𝒏 ∀𝒏 ⇔ 𝒃𝒏 𝐜𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐬

𝜮𝒃𝒏 𝐃𝐢𝐯𝐞𝐫𝐠𝐞𝐬

𝜮𝒃𝒏 𝐂𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐬

Does not show divergence

𝒄𝒏!𝟏 ≤ 𝒄𝒏 ∀𝒏 & 𝐥𝐢𝐦 𝒄𝒏 = 𝟎 𝒏→∞ 𝒂𝒏!𝟏 𝐥𝐢𝐦 < 𝟏 𝒏→∞ 𝒂𝒏

𝑎! !!! ∞

𝑎!

𝐥𝐢𝐦



𝒏→∞

𝐨𝐫 = ∞

Test for Absolute/Conditional Convergence

∞

𝑎! 𝐂𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐬 𝐭𝐡𝐞𝐧 !!!

𝟏 𝒏𝟐

𝒂𝒏!𝟏 >𝟏 𝒏→∞ 𝒂𝒏 𝐨𝐫 = ∞ 𝒏 𝐥𝐢𝐦 𝒂𝒏 > 𝟏



Absolutely Convergent

𝐥𝐢𝐦

𝒏→∞

𝒏

|𝒂𝒏 | < 𝟏

Conditionally Convergent ∞

|𝑎! | Converges !!!

𝒇 𝒙 𝒅𝒙 = 𝒌 𝟏



∞

!!!

∞

𝒑 > 𝟏

∞

𝒂𝒏 is positive and decreasing on [𝟏, ∞)

!!!

∞

Converges 𝒂 𝒓 0 Repeated Solution 𝑏 ! − 4𝑎𝑐 = 0 Complex Solution 𝑥 = 𝛼 ± 𝛽𝑖 if 𝑏 ! − 4𝑎𝑐 < 0

Complete the Square

!

𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐

⇒



log ! 𝑏 =

𝑏 𝑦=𝑎 𝑥+ 2𝑎

!

𝑏! +𝑐− 4𝑎

Other Formulas Distance Formula 𝐷=

Midpoint Formula !

𝑥 − 𝑥!

+ 𝑦 − 𝑦! !

𝑀=

Equation of a Line 𝑠𝑙𝑜𝑝𝑒 = 𝑚 =

𝑥 + 𝑥! 𝑦 + 𝑦! , 2 2

𝑦 = 𝑚𝑥 + 𝑏 𝑦! − 𝑦! = 𝑚 𝑥! − 𝑥! 𝐴𝑥 + 𝐵𝑦 = 𝐶 𝑦 = 𝑎𝑥 ! + 𝑏𝑥 + 𝑐 𝑦 = 𝑎 𝑥 − ℎ ! + 𝑘 ! 𝑥 − ℎ + 𝑦 − 𝑘 ! = 𝑟!

𝑦! − 𝑦! 𝑥! − 𝑥!

Equation of Parabola Vertex: ℎ, 𝑘 Equation of Circle Center: ℎ, 𝑘 Radius: 𝑟 Equation of Ellipse

𝑥−ℎ 𝑎!

!

+

Right Point: ℎ + 𝑎, 𝑘

𝑦−𝑘 𝑏!

!

= 1

Left Point: ℎ − 𝑎, 𝑘 Top Point: ℎ, 𝑘 + 𝑏

Bottom Point: ℎ, 𝑘 − 𝑏 Equation of Hyperbola Center: ℎ, 𝑘 ! Slope: ± !

𝑥−ℎ 𝑎!

!

𝑦−𝑘 − 𝑏!

!

= 1

!

Asymptotes: 𝑦 = ± ! 𝑥 − ℎ + 𝑘 Vertices: ℎ + 𝑎, 𝑘 , ℎ − 𝑎, 𝑘 ! 𝑦−𝑘 𝑥−ℎ − 𝑎! 𝑏!

Equation of Hyperbola Center: ℎ, 𝑘 ! Slope: ± !

!

= 1

!

Asymptotes: 𝑦 = ± ! 𝑥 − ℎ + 𝑘 Vertices: ℎ, 𝑘 + 𝑏 , ℎ, 𝑘 − 𝑏



Areas Square: 𝐴 = 𝐿! = 𝑊 ! Rectangle: 𝐴 = 𝐿 ∙ 𝑊 Circle: 𝐴 = 𝜋 ∙ 𝑟 ! ! ! Ellipse: 𝐴 = 𝜋 ∙ 𝑎𝑏 Triangle: 𝐴 = ! 𝑏 ∙ ℎ Trapezoid: 𝐴 = ! 𝑎 + 𝑏 ∙ ℎ !" Parallelogram: 𝑏 ∙ ℎ Rhombus: 𝐴 = ! , 𝑝 and 𝑞 are the diagonals Surface Areas Cube: 𝐴! = 6𝐿! = 6𝑊 ! Box: 𝐴! = 2(𝐿𝑊 + 𝑊𝐻 + 𝐻𝐿) Sphere: 𝐴! = 4𝜋𝑟 ! Cone: 𝐴! = 𝜋𝑟 𝑟 + ℎ! + 𝑟 ! Cylinder: 2𝜋𝑟ℎ + 2𝜋𝑟 ! Volumes ! Cube: 𝑉 = 𝐿! = 𝑊 ! Box: 𝑉 = 𝐿 ∙ 𝑊 ∙ 𝐻 Sphere: 𝑉 = ! 𝜋 ∙ 𝑟 ! ! ! Cone: 𝑉 = ! 𝜋 ∙ 𝑟 ! ℎ Ellipsoid: 𝑉 = ! 𝜋 ∙ 𝑎𝑏𝑐, 𝑎, 𝑏, 𝑐 are the radii Domain Restrictions 𝑢 𝑦= , 𝑣≠0 𝑦 = 𝑢, 𝑢≥0 𝑦 = ln 𝑢 , 𝑢 > 0 𝑣 ! 𝑦 = 𝑎! , none 𝑦 = 𝑢 none if 𝑛 is odd, 𝑢 ≥ 0 if 𝑛 is even



Right Triangle

𝑥! + 𝑦! = 𝑟! ⇔ cos 𝛼 = tan 𝛼 = sin 𝛼 =

𝑟=

𝑥! + 𝑦!

𝑥 𝑟

𝑦 cos 𝛽 = 𝑟

𝑦 𝑥

𝑥 tan 𝛽 = 𝑦

𝑦 𝑟

𝑥 sin 𝛽 = 𝑟

𝑥 = 𝑟 cos 𝛼 𝑦 = 𝑟 sin 𝛼 𝑦 𝑦 𝛼 = arctan = tan!! 𝑥 𝑥 Reciprocal Identities 1 sin 𝜃 = csc 𝜃 1 csc 𝜃 = sec 𝜃 sin 𝜃 tan 𝜃 = cos 𝜃





𝑦 = 𝑟 cos 𝛽 𝑥 = 𝑟 sin 𝛽 𝛽 = arctan

𝑥 𝑥 = tan!! 𝑦 𝑦

csc 𝜃 =

1 sin 𝜃

tan 𝜃 =

1 cot 𝜃

sec 𝜃 =

1 cos 𝜃

cot 𝜃 =

1 tan 𝜃

cot 𝜃 =

cos 𝜃 sin 𝜃

Double Angle Formulas sin 2𝜃 = 2 sin 𝜃 cos 𝜃 cos 2𝜃 = 1 − 2 sin! 𝜃 cos 2𝜃 = cos ! 𝜃 − sin! 𝜃 cos 2𝜃 = 2 cos ! 𝜃 − 1 ! !"# ! cos 2𝜃 = 1 − 2 sin! 𝜃 tan 2𝜃 = !!!"#! !Officia Half Angle Formulas 1 1 1 − cos(2𝜃) sin! 𝜃 = 1 − cos 2𝜃 cos ! 𝜃 = 1 + 𝑐𝑜𝑠 2𝜃 tan! 𝜃 = 2 2 1 + cos(2𝜃) Sum and Difference Formulas sin 𝛼 ± 𝛽 = sin 𝛼 cos 𝛽 ± cos 𝛼 sin 𝛽 cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 sin 𝛽 tan 𝛼 ± tan 𝛽 tan 𝛼 ± 𝛽 = 1 ∓ tan 𝛼 𝑡𝑎𝑛𝛽 Product to Sum Formulas 1 1 sin 𝛼 sin 𝛽 = [cos 𝛼 − 𝛽 − cos(𝛼 + 𝛽)] cos 𝛼 cos 𝛽 = [cos 𝛼 − 𝛽 + cos(𝛼 + 𝛽)] 2 2 1 1 sin 𝛼 cos 𝛽 = [sin 𝛼 + 𝛽 + sin 𝛼 − 𝛽 ] cos 𝛼 sin 𝛽 = sin 𝛼 + 𝛽 − sin 𝛼 − 𝛽 2 2 Sum to Product Formulas 𝛼+𝛽 𝛼−𝛽 𝛼+𝛽 𝛼−𝛽 sin 𝛼 + sin 𝛽 = 2 sin cos sin 𝛼 − sin 𝛽 = 2 cos sin 2 2 2 2 𝛼+𝛽 𝛼−𝛽 𝛼+𝛽 𝛼−𝛽 cos 𝛼 + cos 𝛽 = 2 cos cos cos 𝛼 − cos 𝛽 = −2 sin sin 2 2 2 2



Unit Circle





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