Calculus 2 Formula Sheet - Chapter 7 - u-substitution Chain Rule and Integration Formulas

Oct 13, 2026

Upload by Ezra

Calculus 2 formula sheet compiling key results across Chapter 7 and beyond. It includes chain rule derivatives, u-substitution integration templates, and trigonometric and logarithmic forms for common transformations. The sheet further lists integration techniques such as integration by parts, trigonometric substitution cases, and arc length and surface area formulas. It also summarizes parametric curve calculus, polar curve relationships, tangent classification, and area formulas between curves and in polar coordinates.

LanguageEnglish

CategoryPaper Templates

FormatPDF

Pages10

Reading Time4 min

AccessFree to download

Chapter 7

Chain Rule and u-substitution

Integration by Parts

Trigonometric Substitution Cases

Arc Length and Surface Area

Derivatives and Parametric/POLAR Formulas

Derivative and Second Derivative

Area and Arc Length

Parametric and Polar Curve Calculus

Calculus 2 Formula Sheet Chapter 7 􀀡􀀡􀀢 (􀀤 ! ) = 􀀤 ! Chain Rule ""! (􀀤 \# ) = 􀀤 \# ""\#! ' 􀀤 ! 􀀡􀀢 = 􀀤 ! + 􀀩 u-substitution ∫ 􀀤 \# 􀀡􀀢 = 􀀤 \# + 􀀩 ""! + 􀀬􀀭 (􀀢 ). = $! for 􀀢 > 0 Chain Rule ""! + 􀀬􀀭 (􀀱 ). = $\# ⋅ ""\#!""! (􀀬􀀭|􀀢 |) = $! for 􀀢 ≠ 0 ' 1􀀢 􀀡􀀢 = 􀀬􀀭 (|􀀢 |) + 􀀶 u-substitution ∫ $\# 􀀡􀀱 = 􀀬􀀭 (|􀀱 |) + 􀀶 ' tan (􀀢 )􀀡􀀢 = 􀀬􀀭 (|sec (􀀢 ) |) + 􀀶' cot (􀀢 )􀀡􀀢 = 􀀬􀀭 (|sin(􀀢 ) |) + 􀀶 | 􀀡􀀡􀀢 +􀀬􀁀􀁁%(􀀢 ). = ln~~ ~~(1􀁃)~~ ~~􀀢 Chain Rule ""! +􀀬􀁀􀁁%(􀀱 ). = &'~~ ~~($%)! ⋅ ""\#!
􀀡􀀡􀀢 (􀁃 ! ) = ln (􀁃)􀁃 ! | | | | | --- | --- | --- | --- | | Chain Rule ""!
' 􀁃 ! 􀀡􀀢
u-substitution | (􀁃\# ) = ln (􀁃)􀁃 \# ⋅ ""\#! | | | | | = | 􀁃 !
ln (􀁃) | + 􀀶 | | | ∫ 􀁃\# 􀀡􀀢 = &'%(!%) + 􀀶 | | | 􀀡􀀡􀀢 +􀁄􀁅􀀭 + $ (􀀢 ). = √1~~ ~~~~ ~~􀀢~~ ,~~ Chain Rule ""! +􀁄􀁅􀀭 + $ (􀀱 ). = √$~~ ~~\#" ⋅ ""\#! 1 ' ~~ ~~ 􀀡􀀢 = 􀁄􀁅􀀭 + $ (􀀢 ) + 􀀶 √1 − 􀀢 , u-substitution ∫ √$~~ ~~\#" 􀀡􀀢 = 􀁄􀁅􀀭 + $ (􀀱) + 􀀶 􀀡􀀡􀀢 + 􀀩􀁀􀁄 + $ (􀀢 ). = − √1~~ ~~~~ ~~􀀢~~ ,~~ Chain Rule ""! + 􀀩􀁀􀁄 + $ (􀀱 ). = − √$~~ ~~\#" ⋅ ""\#! 􀀡􀀡􀀢 + 􀁈􀁉􀀭 + $ (􀀢 ). = 1~~ ~~􀀢~~ , ~~Chain Rule ""! + 􀁈􀁉􀀭 + $ (􀀱). = $~~ ~~$.\#" ⋅ ""\#! ' ~~ ~~1~~ ~~􀀢~~ ,~~ 􀀡􀀢 = 􀁈􀁉􀀭 + $ (􀀢 ) + 􀀶 u-substitution ∫ $~~ ~~$.\#" 􀀡􀀢 = 􀁈􀁉􀀭 + $ (􀀱 ) + 􀀶 􀁉~~ , ~~~~ ~~􀀢~~ ,~~ 􀀡􀀢 = 1􀁉 􀁈􀁉􀀭 + $ J􀀢􀁉K + 􀀶 ' ' √􀁉~~ , ~~~~ ~~􀀢~~ ,~~ 􀀡􀀢 = 􀁄􀁅􀀭 + $ J􀀢􀁉K + 􀀶 􀀡􀀡􀀢 +􀁄􀀤􀀩 + $ (􀀢 ). = 􀀢~~ ~~√􀀢~~ ~~1~~, ~~−~~ ~~1 Chain Rule ""! +􀁄􀀤􀀩 + $ (􀀱 ). = \#√$\#"~~ + ~~$ ⋅ ""\#! 1 ' ~~ ~~ 􀀡􀀢 = 􀁄􀀤􀀩 + $ (􀀢 ) + 􀀶 􀀢 √􀀢 , − 1 u-substitution ∫ \#√$\#"~~ + ~~$ 􀀡􀀢 = 􀁄􀀤􀀩 + $ (􀀱 ) + 􀀶 Chapter 8 Integration By Parts ' 􀀱􀀡􀁌 = 􀀱􀁌 − ' 􀁌􀀡􀀱 ' sec (􀀱 ) 􀀡􀀱 = 􀀬􀀭|sec (􀀱 ) + tan (􀀱 ) | + 􀀩' csc (􀀱 ) 􀀡􀀱 = −􀀬􀀭|csc (􀀱 ) + cot(􀀱 ) | + 􀀩' tan (􀀱 ) 􀀡􀀱 = 􀀬􀀭|􀁄􀀤􀀩 (􀀱 ) | + 􀀩' 􀀩􀁀􀁈 (􀀱 ) 􀀡􀀱 = 􀀬􀀭|􀁄􀁅􀀭 (􀀱) | + 􀀩 ' 􀁄􀀤􀀩 / (􀀱 )􀀡􀀱 = 12 [sec (􀀱) tan (􀀱 ) + 􀀬􀀭|sec (􀀱 ) + tan (􀀱 ) |] + 􀀩 Trigonometric Substitution Case 1: √􀁐 􀀰 − 􀁑 􀀰 where 􀁐 > 􀁒 􀀢 = 􀁉􀁄􀁅􀀭 (􀁓) 􀀡􀀢 = 􀁉􀀩􀁀􀁄 (􀁓) T􀁉 , − 􀀢 , = 􀁉􀀩􀁀􀁄 (􀁓) Case 2: √􀁐 􀀰 + 􀁑 􀀰 where 􀁐 > 􀁒 􀀢 = 􀁉􀁈􀁉􀀭 (􀁓) 􀀡􀀢 = 􀁉􀁄􀀤􀀩 ,(􀁓) T􀁉 , − 􀀢 , = 􀁉􀁄􀀤􀀩 (􀁓) Case 3: √􀁑 􀀰 − 􀁐 􀀰 where 􀁐 > 􀁒 􀀢 = 􀁉􀁄􀀤􀀩 (􀁓) 􀀡􀀢 = 􀁉􀁄􀀤􀀩 (􀁓)􀁈􀁉􀀭 (􀁓) T􀀢 , − 􀁉 , = 􀁉􀁈􀁉􀀭 (􀁓) Chapter 9 Arc Length Formulas 􀀡 = ∫ $1~~ ~~+~~ ~~[~~ ~~􀀩for~~´~~􀀡(􀀬=)]􀀣~~!~~(􀀥􀀬ovoerr􀀡􀀧 01 + 1$$%&2! 􀀯􀀬 􀀡 = ∫'$ $1~~ ~~+~~ ~~[~~ ~~􀀳~~´ ~~(􀀴)]~~!~~ 􀀯􀀴 or 􀀡 = ∫'$ 01 + 1$$&%2! 􀀯􀀴 for 􀀥 = 􀀪 (􀀡) over 􀀫 ≤ 􀀡 ≤ 􀀬 Surface Area Formulas 􀀵􀀶 = 7 􀀸􀀹􀀺􀀻􀀼 Revolve about the x-axis 􀁕􀁖 = ∫ 􀁗􀁘􀁙 􀁚􀁛 where 􀀡􀁄 = \ 1 + ]~~􀀨~~􀀨􀀩􀀪^ 2 􀀯􀀬 􀀽􀀾 􀀡􀁄 = _ 1 + ]~~􀀨~~􀀨􀀪􀀩^ 2 􀀯􀀴 Revolve about the y-axis 􀀵􀀶 = ∫ 􀀸􀀹􀀺􀀿􀀻􀀼 where 􀀡􀁄 = \ 1 + ]~~􀀨~~􀀨􀀩􀀪^ 2 􀀯􀀬 􀀽􀀾 􀀡􀁄 = _ 1 + ]~~􀀨~~􀀨􀀪􀀩^2 􀀯􀀴 Parametric Equation Calculus Formulas 􀀢 = 􀁠 (􀁈) 􀁡 = 􀁁 (􀁈) 􀁈 ∈ 􀁣 Derivative !!"\# = ~~ ~~"~~\#~~$~~ ~~ assuming $$&, ≠ 0 !\# Second Derivative 􀀯 !􀀴 ~~ ~~􀀯􀀯􀁂 C 􀀯􀀯􀀴􀀬D = 􀀯􀀬 ! 􀀯􀀬􀀯􀁂 Area 􀁅 = 7&& " 􀀴􀀯􀀬 = 7,, /􀀳 (􀁂)􀀩 - (􀁂)􀀯􀁂 Arc Length 􀁆 = 70/ GC~~ ~~􀀯􀀯􀀬􀁂D! + C~~ ~~􀀯􀀯􀀴􀁂D! 􀀯􀁂 􀁈 ≤ 􀁂 ≤ 􀁊 Surface Area 􀁕􀁖 = ' 􀁗􀁘􀁤􀁚􀁛 Rotate a Parametric Curve C about the x-axis 􀁥􀁦 = '􀀲􀀱 2􀁌􀀾_g􀀯􀀯􀀬􀁂h2 + g~~ ~~􀀯􀀯􀀴􀁂h 2 􀀯􀁂 = '􀀲􀀱 2􀁌􀀴_g􀀯􀀯􀀬􀁂h 2 + g~~ ~~􀀯􀀯􀀴􀁂h2 􀀯􀁂 Rotate a Parametric Curve C about the y-axis 􀁥􀁦 = '􀀲􀀱 2􀀾_g􀀯􀀯􀀬􀁂h2 + g~~ ~~􀀯􀀯􀀴􀁂h 2 􀀯􀁂 = '􀀲􀀱 2􀁌􀀬_g􀀯􀀯􀀬􀁂h 2 + g~~ ~~􀀯􀀯􀀴􀁂h 2 􀀯􀁂 Calculus of Polar Curves 􀀭 = 􀀣 (􀀮) | $%
$& | = | !"
!\#
!$

!\# | = | !"
!%
!$

!% | = | !!% 34567(9);
!!% 34'<5 (9); | = | !!&%567 (9)=4'<5 (9)
!!&%'<5 (9)>4567 (9) | if | $$&9 ≠ 0 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | Horizontal Tangents Points on the curve in which $$%9 = 0, but $$&9 ≠ 0 Vertical Tangents Points on the curve in which $$&9 = 0, but $$%9 ≠ 0 Tangent Line at the Pole "3 $4 We let 􀀭 = 0 and ~~ ~~ = 􀁈􀁉􀀭 (􀁓) if ≠ 0 "! $9 Area 􀁩 = 􀁠 (􀁓) 􀁪 ≤ 􀁓 ≤ 􀁬 􀁦 = ∫ $,[􀁠(􀁓)], 􀀡􀁓 4 1 􀁦 = ' 2 􀁩 , 􀀡􀁓 5 Area Between Two Curves 􀁩 = 􀁠 (􀁓) 􀁉􀀭􀀡 􀁩 = 􀁁 (􀁓) 􀁪 ≤ 􀁓 ≤ 􀁬 1 4 􀁦 = 2 ' (􀁩, 6 − 􀁩,%) 􀀡􀁓 5 Arc Length 􀁩 = 􀁠 (􀁓) 􀁪 ≤ 􀁓 ≤ 􀁬 􀁥 = '54 _m􀀬􀀬􀀭􀀮n2 + 􀀭2 􀀬􀀮

FAQs

What topics are covered in the Calculus 2 formula sheet?

It covers chain rule derivatives, u-substitution integration, integration by parts, trigonometric substitution, and formulas for arc length, surface area, and area problems, including parametric and polar curve relationships.

How is u-substitution used in the sheet?

The sheet provides substitution-based integral templates showing how integrals can be rewritten to match a derivative inside a composition, turning them into simpler forms.

What are the common trigonometric substitution cases included?

It lists case-based rules for radicals of forms like sqrt(a^2-b^2) and sqrt(a^2+b^2), assigning specific substitutions and corresponding differentials to rewrite the integral.

Oct 13, 2026

Upload by Ezra