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Common Derivatives
๐‘‘
๐‘ฅ =1
๐‘‘๐‘ฅ
๐‘‘
๐‘‘
[๐‘Ž๐‘“ ๐‘ฅ ] = ๐‘Ž [๐‘“ ๐‘ฅ ]
๐‘‘๐‘ฅ
๐‘‘๐‘ฅ
Limits & Derivatives Cheat Sheet
๐‘‘
๐‘Ž๐‘ฅ = ๐‘Ž
๐‘‘๐‘ฅ
๐‘‘
๐‘Ž๐‘ฅ ๐‘› = ๐‘›๐‘Ž๐‘ฅ ๐‘›−1
๐‘‘๐‘ฅ
Properties of Limits
๐‘‘
๐‘ =0
๐‘‘๐‘ฅ
๐‘‘
[๐‘“ ๐‘ฅ ]๐‘› = ๐‘›[๐‘“ ๐‘ฅ ]๐‘›−1 ๐‘“′(๐‘ฅ)
๐‘‘๐‘ฅ
lim ๐‘๐‘“ ๐‘ฅ
= ๐‘ lim ๐‘“(๐‘ฅ)
๐‘ฅ→๐‘Ž
๐‘ฅ→๐‘Ž
๐‘‘ 1
= −๐‘›๐‘ฅ −
๐‘‘๐‘ฅ ๐‘ฅ ๐‘›
lim [๐‘“ ๐‘ฅ ± ๐‘” ๐‘ฅ ] = lim ๐‘“(๐‘ฅ) ± lim ๐‘”(๐‘ฅ)
๐‘ฅ→๐‘Ž
๐‘ฅ→๐‘Ž
๐‘ฅ→๐‘Ž
lim [๐‘“ ๐‘ฅ ๐‘” ๐‘ฅ ] = lim ๐‘“(๐‘ฅ) lim ๐‘”(๐‘ฅ)
๐‘ฅ→๐‘Ž
lim
๐‘ฅ→๐‘Ž
๐‘ฅ→๐‘Ž
๐‘ฅ→๐‘Ž
lim [๐‘“ ๐‘ฅ ]๐‘› = lim ๐‘“ ๐‘ฅ
๐‘ฅ→๐‘Ž
๐‘›
๐‘ฅ→๐‘Ž
Limit Evaluations At ±∞
lim ๐‘’ ๐‘ฅ = ∞ and lim ๐‘’ ๐‘ฅ = 0
๐‘ฅ→+∞
๐‘ฅ→−∞
lim ln ๐‘ฅ = ∞ and lim+ ln ๐‘ฅ = −∞
๐‘ฅ→∞
๐‘ฅ→0
๐‘
if r > 0: lim ๐‘Ÿ = 0
๐‘ฅ→∞ ๐‘ฅ
๐‘
if r > 0 & ∀๐‘ฅ > 0 ๐‘ฅ ๐‘Ÿ ∈ โ„ โˆถ lim ๐‘Ÿ = 0
๐‘ฅ→−∞ ๐‘ฅ
lim ๐‘ฅ ๐‘Ÿ = ∞ for even r
๐‘ฅ→±∞
lim ๐‘ฅ ๐‘Ÿ = ∞ and lim ๐‘ฅ ๐‘Ÿ = −∞ for odd r
๐‘ฅ→+∞
๐‘ฅ→−∞
L’Hopital’s Rule
๐‘“(๐‘ฅ) 0
±∞
๐‘“(๐‘ฅ)
๐‘“′(๐‘ฅ)
If lim
= or
then lim
= lim
๐‘ฅ→๐‘Ž ๐‘”(๐‘ฅ)
๐‘ฅ→๐‘Ž ๐‘”(๐‘ฅ)
๐‘ฅ→๐‘Ž ๐‘”′(๐‘ฅ)
0
±∞
Derivative Definition
๐‘‘
๐‘“ ๐‘ฅ
๐‘‘๐‘ฅ
= ๐‘“ ′ ๐‘ฅ = lim
โ„Ž→0
๐‘“ ๐‘ฅ + โ„Ž − ๐‘“(๐‘ฅ)
โ„Ž
Product Rule
๐‘“ ๐‘ฅ ๐‘” ๐‘ฅ
′
= ๐‘“ ′ ๐‘ฅ ๐‘” ๐‘ฅ + ๐‘“ ๐‘ฅ ๐‘”′(๐‘ฅ)
Quotient Rule
๐‘‘ ๐‘“ ๐‘ฅ
๐‘‘๐‘ฅ ๐‘” ๐‘ฅ
=
๐‘“ ′ ๐‘ฅ ๐‘” ๐‘ฅ − ๐‘“ ๐‘ฅ ๐‘”′(๐‘ฅ)
[๐‘” ๐‘ฅ ]2
Chain Rule
๐‘‘
๐‘“ ๐‘” ๐‘ฅ
๐‘‘๐‘ฅ
=
๐‘“′
๐‘” ๐‘ฅ ๐‘”′(๐‘ฅ)
Basic Properties of Derivatives
๐‘๐‘“ ๐‘ฅ
๐‘“ ๐‘ฅ ±๐‘” ๐‘ฅ
′
= ๐‘[๐‘“ ′ ๐‘ฅ ]
′
= ๐‘“′(๐‘ฅ) ± ๐‘”′(๐‘ฅ)
=−
๐‘›
๐‘ฅ ๐‘›+1
Derivatives of Trigonometric Functions
๐‘ฅ→๐‘Ž
lim ๐‘“(๐‘ฅ)
= ๐‘ฅ→๐‘Ž
๐‘–๐‘“ lim ๐‘”(๐‘ฅ) ≠ 0
๐‘ฅ→๐‘Ž
lim ๐‘”(๐‘ฅ)
๐‘“ ๐‘ฅ
๐‘” ๐‘ฅ
๐‘›+1
๐‘‘
sin ๐‘ฅ = cos ๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘
cos ๐‘ฅ = − sin ๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘
tan ๐‘ฅ = sec 2 ๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘
sec ๐‘ฅ = sec ๐‘ฅ tan ๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘
csc ๐‘ฅ = − csc ๐‘ฅ cot ๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘
cot ๐‘ฅ = −csc 2 ๐‘ฅ
๐‘‘๐‘ฅ
Derivatives of Exponential & Logarithmic Functions
๐‘‘ ๐‘ฅ
๐‘’ = ๐‘’๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘ ๐‘ฅ
๐‘Ž = ๐‘Ž ๐‘ฅ ln ๐‘Ž
๐‘‘๐‘ฅ
๐‘‘
1
ln |๐‘ฅ| =
๐‘‘๐‘ฅ
๐‘ฅ
๐‘‘
1
ln ๐‘ฅ = , ๐‘ฅ > 0
๐‘‘๐‘ฅ
๐‘ฅ
๐‘‘
1
log๐‘Ž ๐‘ฅ =
๐‘‘๐‘ฅ
๐‘ฅ ln ๐‘Ž
๐‘‘ ๐‘“(๐‘ฅ)
๐‘’
= ๐‘“′(๐‘ฅ)๐‘’ ๐‘“(๐‘ฅ)
๐‘‘๐‘ฅ
๐‘‘
๐‘“ ๐‘ฅ
๐‘‘๐‘ฅ
๐‘” ๐‘ฅ
=๐‘“ ๐‘ฅ
๐‘” ๐‘ฅ
๐‘‘
๐‘“′(๐‘ฅ)
ln ๐‘“(๐‘ฅ) =
๐‘‘๐‘ฅ
๐‘“(๐‘ฅ)
๐‘‘ ๐‘“
๐‘Ž
๐‘‘๐‘ฅ
๐‘ฅ
= ๐‘Ž ๐‘“(๐‘ฅ) ln ๐‘Ž ๐‘“′(๐‘ฅ)
๐‘” ๐‘ฅ ๐‘“′ ๐‘ฅ
+ ln ๐‘“ ๐‘ฅ
๐‘“ ๐‘ฅ
๐‘”′ ๐‘ฅ
Derivatives of Inverse Trig Functions
๐‘‘
1
sin−1 ๐‘ฅ =
๐‘‘๐‘ฅ
1 − ๐‘ฅ2
๐‘‘
1
sec −1 ๐‘ฅ =
๐‘‘๐‘ฅ
|๐‘ฅ| ๐‘ฅ 2 − 1
๐‘‘
1
cos−1 ๐‘ฅ = −
๐‘‘๐‘ฅ
1 − ๐‘ฅ2
๐‘‘
1
csc −1 ๐‘ฅ = −
๐‘‘๐‘ฅ
|๐‘ฅ| ๐‘ฅ 2 − 1
๐‘‘
1
tan−1 ๐‘ฅ =
๐‘‘๐‘ฅ
1 + ๐‘ฅ2
๐‘‘
1
cot−1 ๐‘ฅ = −
๐‘‘๐‘ฅ
1 + ๐‘ฅ2
Derivatives of Hyperbolic Functions
๐‘‘
sinh ๐‘ฅ = cosh ๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘
sech ๐‘ฅ = − coth ๐‘ฅ csch ๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘
cosh ๐‘ฅ = sinh ๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘
csch ๐‘ฅ = − tanh ๐‘ฅ sech ๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘
tanh ๐‘ฅ = 1 − tanh2 ๐‘ฅ
๐‘‘๐‘ฅ
๐‘‘
coth ๐‘ฅ = −1 − coth2 ๐‘ฅ
๐‘‘๐‘ฅ
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Integrals of Trigonometric Functions
เถฑ cos ๐‘ฅ ๐‘‘๐‘ฅ = sin ๐‘ฅ + ๐ถ
เถฑ csc ๐‘ฅ cot ๐‘ฅ ๐‘‘๐‘ฅ = − csc ๐‘ฅ + ๐ถ
เถฑ sin ๐‘ฅ ๐‘‘๐‘ฅ = −cos ๐‘ฅ + ๐ถ
เถฑ sec ๐‘ฅ tan ๐‘ฅ ๐‘‘๐‘ฅ = sec ๐‘ฅ + ๐ถ
เถฑ sec 2 ๐‘ฅ ๐‘‘๐‘ฅ = tan ๐‘ฅ + ๐ถ
เถฑ csc 2 ๐‘ฅ ๐‘‘๐‘ฅ = − cot ๐‘ฅ + ๐ถ
เถฑ cot ๐‘ฅ ๐‘‘๐‘ฅ = ln | sin ๐‘ฅ | + ๐ถ
เถฑ csc ๐‘ฅ ๐‘‘๐‘ฅ = ln | csc ๐‘ฅ − cot ๐‘ฅ | + ๐ถ
เถฑ sinh ๐‘ฅ ๐‘‘๐‘ฅ = cosh ๐‘ฅ + ๐ถ
เถฑ cosh ๐‘ฅ ๐‘‘๐‘ฅ = sinh ๐‘ฅ + ๐ถ
Integration Cheat Sheet
Integration Properties
๐‘
๐‘
เถฑ ๐‘๐‘“ ๐‘ฅ ๐‘‘๐‘ฅ = ๐‘ เถฑ ๐‘“ ๐‘ฅ ๐‘‘๐‘ฅ
๐‘Ž
๐‘Ž
๐‘
๐‘
๐‘
เถฑ ๐‘“ ๐‘ฅ ± ๐‘” ๐‘ฅ ๐‘‘๐‘ฅ = เถฑ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ ± เถฑ ๐‘”(๐‘ฅ) ๐‘‘๐‘ฅ
๐‘Ž
๐‘Ž
๐‘Ž
๐‘Ž
เถฑ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ = 0
๐‘Ž
๐‘
๐‘Ž
เถฑ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ = − เถฑ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ
๐‘Ž
๐‘
๐‘
๐‘
เถฑ tan ๐‘ฅ ๐‘‘๐‘ฅ = ln | sec ๐‘ฅ | + ๐ถ
๐‘
เถฑ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ + เถฑ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ = เถฑ ๐‘“ ๐‘ฅ ๐‘‘๐‘ฅ
๐‘Ž
๐‘
๐‘Ž
Fundamental Theorem of Calculus
เถฑ sec ๐‘ฅ ๐‘‘๐‘ฅ = ln | sec ๐‘ฅ + tan ๐‘ฅ | + ๐ถ
๐‘
เถฑ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ = [๐น ๐‘ฅ ]๐‘๐‘Ž = ๐น ๐‘ − ๐น ๐‘Ž
๐‘Ž
เถฑ
where ๐‘“ is continuous on ๐‘Ž, ๐‘ and ๐‘“ ′ = ๐น
Definite Integral Definition
เถฑ
1
1
๐‘ฅ
๐‘‘๐‘ฅ = tan−1
+๐ถ
๐‘Ž2 + ๐‘ฅ 2
๐‘Ž
๐‘Ž
1
๐‘Ž2
−
๐‘ฅ2
๐‘‘๐‘ฅ = sin−1
๐‘ฅ
+๐ถ
๐‘Ž
๐‘›
เถฑ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ = lim เท ๐‘“ ๐‘ฅ๐‘˜ โˆ†๐‘ฅ
๐‘›→∞
๐‘Ž
where โˆ†๐‘ฅ =
Partial Fractions
๐‘˜=1
๐‘−๐‘Ž
๐‘Ž๐‘›๐‘‘ ๐‘ฅ๐‘˜ = ๐‘Ž + ๐‘˜โˆ†๐‘ฅ
๐‘›
Common Integrals
เถฑ ๐‘˜ ๐‘‘๐‘ฅ = ๐‘˜๐‘ฅ + ๐ถ
1
เถฑ ๐‘‘๐‘ฅ = ln |๐‘ฅ| + ๐ถ
๐‘ฅ
เถฑ ๐‘’ ๐‘ฅ ๐‘‘๐‘ฅ = ๐‘’ ๐‘ฅ + ๐ถ
เถฑ
เถฑ ln ๐‘ฅ ๐‘‘๐‘ฅ = ๐‘ฅ ln ๐‘ฅ − ๐‘ฅ + ๐ถ
เถฑ ๐‘ ๐‘ฅ ๐‘‘๐‘ฅ =
๐‘๐‘ฅ
+๐ถ
ln ๐‘
1
1
๐‘‘๐‘ฅ = ln |๐‘Ž๐‘ฅ + ๐‘| + ๐ถ
๐‘Ž๐‘ฅ + ๐‘
๐‘Ž
เถฑ
เถฑ
๐‘ฅ ๐‘›+1
เถฑ ๐‘ฅ ๐‘› ๐‘‘๐‘ฅ =
+๐ถ
๐‘›+1
1
1
๐‘ฅ−๐‘Ž
๐‘‘๐‘ฅ =
ln
2
2
๐‘ฅ −๐‘Ž
2๐‘Ž
๐‘ฅ+๐‘Ž
1
๐‘ฅ 2 ± ๐‘Ž2
๐‘…(๐‘ฅ)
๐ด1
๐ด2
๐ด๐‘›
=
+
+ โ‹ฏ+
,
๐‘„(๐‘ฅ)
๐‘Ž1 ๐‘ฅ + ๐‘1
๐‘Ž2 ๐‘ฅ + ๐‘2
(๐‘Ž๐‘› ๐‘ฅ + ๐‘๐‘› )
where ๐‘„ ๐‘ฅ = ๐‘Ž1 ๐‘ฅ + ๐‘1 ๐‘Ž2 ๐‘ฅ + ๐‘2 … (๐‘Ž๐‘› ๐‘ฅ + ๐‘๐‘› )
๐‘…(๐‘ฅ)
๐ด1
๐ด2
๐ด๐‘›
=
+
+โ‹ฏ+
2
๐‘„(๐‘ฅ) ๐‘Ž1 ๐‘ฅ + ๐‘1 (๐‘Ž1 ๐‘ฅ + ๐‘1 )
๐‘Ž1 ๐‘ฅ + ๐‘1
where a linear factor of ๐‘„ ๐‘ฅ is repeated ๐‘› times
๐‘›
๐‘…(๐‘ฅ)
๐ด๐‘ฅ + ๐ต
= 2
๐‘„(๐‘ฅ) ๐‘Ž๐‘ฅ + ๐‘๐‘ฅ + ๐‘
where ๐‘„ ๐‘ฅ has a factor ๐‘Ž๐‘ฅ 2 + ๐‘๐‘ฅ + ๐‘, where ๐‘ 2 − 4๐‘Ž๐‘ < 0
๐‘…(๐‘ฅ)
๐ด1 ๐‘ฅ + ๐ต1
๐ด2 ๐‘ฅ + ๐ต2
๐ด๐‘› ๐‘ฅ + ๐ต๐‘›
= 2
+
+ โ‹ฏ+
๐‘„ ๐‘ฅ
๐‘Ž๐‘ฅ + ๐‘๐‘ฅ + ๐‘ (๐‘Ž๐‘ฅ 2 + ๐‘๐‘ฅ + ๐‘)2
๐‘Ž๐‘ฅ 2 + ๐‘๐‘ฅ + ๐‘
where ๐‘„ ๐‘ฅ has a factor
๐‘Ž๐‘ฅ 2
+ ๐‘๐‘ฅ + ๐‘ , where
๐‘2
− 4๐‘Ž๐‘ < 0
Integration by Parts
เถฑ ๐‘ข ๐‘‘๐‘ฃ = ๐‘ข๐‘ฃ − เถฑ ๐‘ฃ ๐‘‘๐‘ข,
๐‘›
@SmartGirlStudy
www.SmartGirlStudy.com
๐‘
where ๐‘ฃ = เถฑ ๐‘‘๐‘ฃ
or เถฑ ๐‘“ ๐‘ฅ ๐‘”′(๐‘ฅ) ๐‘‘๐‘ฅ = ๐‘“ ๐‘ฅ ๐‘” ๐‘ฅ − เถฑ ๐‘“ ′ ๐‘ฅ ๐‘”(๐‘ฅ) ๐‘‘๐‘ฅ
๐‘‘๐‘ฅ = ln ๐‘ฅ + ๐‘ฅ 2 ± ๐‘Ž2
Integrals of Symmetric Functions
Integration by Substitution
๐‘
๐‘”(๐‘)
เถฑ ๐‘“ ๐‘” ๐‘ฅ ๐‘”′(๐‘ฅ) ๐‘‘๐‘ฅ = เถฑ
๐‘Ž
๐‘”(๐‘Ž)
๐‘“ ๐‘ข ๐‘‘๐‘ข
where ๐‘ข = ๐‘” ๐‘ฅ and ๐‘‘๐‘ข = ๐‘”′ ๐‘ฅ ๐‘‘๐‘ฅ
๐‘Ž
๐‘Ž
If ๐‘“ is even ๐‘“ −๐‘ฅ = ๐‘“ ๐‘ฅ , then เถฑ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ = 2 เถฑ ๐‘“ ๐‘ฅ ๐‘‘๐‘ฅ
−๐‘Ž
0
๐‘Ž
If ๐‘“ is odd ๐‘“ −๐‘ฅ = −๐‘“ ๐‘ฅ , then เถฑ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ = 0
−๐‘Ž
Applications of Integration
๐‘
Area Between 2 Curves
Integration & Multivariate Calculus
Cheat Sheet
๐ด = เถฑ |๐‘“ ๐‘ฅ − ๐‘” ๐‘ฅ | ๐‘‘๐‘ฅ
๐‘Ž
๐‘›
๐‘
lim เท ๐ด(๐‘ฅ๐‘–∗ )โˆ†๐‘ฅ = เถฑ ๐ด(๐‘ฅ) ๐‘‘๐‘ฅ
Volume Definition
๐‘›→∞
๐‘Ž
๐‘–=1
Integration by Trigonometric Substitution
Expression
Substitution
Evaluation
Identity Used
๐‘Ž2 − ๐‘ฅ 2
๐‘ฅ = ๐‘Ž sin ๐œƒ
๐‘‘๐‘ฅ = ๐‘Ž cos ๐œƒ ๐‘‘๐œƒ
๐‘Ž2 − ๐‘Ž2 sin2 ๐œƒ = ๐‘Ž cos ๐œƒ
1 − sin2 ๐œƒ = cos 2 ๐œƒ
๐‘ฅ 2 − ๐‘Ž2
๐‘ฅ = ๐‘Ž sec ๐œƒ
๐‘‘๐‘ฅ = ๐‘Ž sec ๐œƒ tan ๐œƒ ๐‘‘๐œƒ
๐‘Ž2 sec 2 ๐œƒ − ๐‘Ž2 = ๐‘Ž tan ๐œƒ
sec2 ๐œƒ − 1 = tan2 ๐œƒ
๐‘Ž2 + ๐‘ฅ 2
๐‘ฅ = ๐‘Ž tan ๐œƒ
๐‘‘๐‘ฅ = ๐‘Ž sec2 ๐œƒ ๐‘‘๐œƒ
๐‘Ž2 + ๐‘Ž2 tan2 ๐œƒ = ๐‘Ž sec ๐œƒ
1 + tan2 ๐œƒ = sec 2 ๐œƒ
Strategy for Evaluating โ€ซ๐’™๐’… ๐’™ ๐’๐ฌ๐จ๐œ ๐’™ ๐’Ž๐ง๐ข๐ฌ ืฌโ€ฌ
Strategy for Evaluating โ€ซ๐’™๐’… ๐’™ ๐’ ๐œ๐ž๐ฌ ๐’™ ๐’Ž๐ง๐š๐ญ ืฌโ€ฌ
If ๐‘› is odd
If ๐‘› is even
เถฑ sin๐‘š ๐‘ฅ cos 2๐‘˜+1 ๐‘ฅ ๐‘‘๐‘ฅ = เถฑ sin๐‘š ๐‘ฅ cos2 ๐‘ฅ
= เถฑ sin๐‘š ๐‘ฅ 1 − sin2 ๐‘ฅ
๐‘˜
๐‘˜ cos ๐‘ฅ ๐‘‘๐‘ฅ
เถฑ tan๐‘š ๐‘ฅ sec 2๐‘˜ ๐‘ฅ ๐‘‘๐‘ฅ = เถฑ tan๐‘š ๐‘ฅ sec 2 ๐‘ฅ
= เถฑ tan๐‘š ๐‘ฅ 1 + tan2 ๐‘ฅ
cos ๐‘ฅ ๐‘‘๐‘ฅ
Then substitute: ๐‘ข = sin ๐‘ฅ
เถฑ sin2๐‘˜+1 ๐‘ฅ cos ๐‘› ๐‘ฅ ๐‘‘๐‘ฅ = เถฑ sin2 ๐‘ฅ
๐‘˜
If ๐‘š is odd
๐‘˜ cos ๐‘› ๐‘ฅ sin ๐‘ฅ ๐‘‘๐‘ฅ
เถฑ tan2๐‘˜+1 ๐‘ฅ sec ๐‘› ๐‘ฅ ๐‘‘๐‘ฅ
cos ๐‘› ๐‘ฅ sin ๐‘ฅ ๐‘‘๐‘ฅ
= เถฑ tan2 ๐‘ฅ
Then substitute: ๐‘ข = cos ๐‘ฅ
๐‘‘
๐‘“ ๐‘ก ๐ฎ๐‘ก
๐‘‘๐‘ก
๐‘‘
๐ฎ ๐‘ก โˆ™๐ฏ ๐‘ก
๐‘‘๐‘ก
๐‘‘
๐ฎ ๐‘ก ×๐ฏ ๐‘ก
๐‘‘๐‘ก
๐‘‘
๐ฎ ๐‘“ ๐‘ก
๐‘‘๐‘ก
sec ๐‘›−1 ๐‘ฅ sec ๐‘ฅ tan ๐‘ฅ ๐‘‘๐‘ฅ
Then substitute: ๐‘ข = sec ๐‘ฅ
Product Identities
เถฑ sin ๐‘š๐‘ฅ cos ๐‘›๐‘ฅ ๐‘‘๐‘ฅ
sin ๐ด cos ๐ต
1
= sin ๐ด − ๐ต + sin ๐ด + ๐ต
2
เถฑ sin ๐‘š๐‘ฅ sin ๐‘›๐‘ฅ ๐‘‘๐‘ฅ
sin ๐ด sin ๐ต
1
= cos ๐‘Ž − ๐ต − cos ๐ด + ๐ต
2
เถฑ cos ๐‘š๐‘ฅ cos ๐‘›๐‘ฅ ๐‘‘๐‘ฅ
cos ๐ด cos ๐ต
1
= cos ๐ด − ๐ต + cos ๐ด + ๐ต
2
Derivatives of Vector Functions
๐‘‘
๐‘๐ฎ ๐‘ก
๐‘‘๐‘ก
๐‘˜
= เถฑ sec 2 ๐‘ฅ − 1 sec ๐‘›−1 ๐‘ฅ sec ๐‘ฅ tan ๐‘ฅ ๐‘‘๐‘ฅ
If ๐‘› and ๐‘š are even
Use the half angle identities:
1
1
sin2 ๐‘ฅ = 1 − cos 2๐‘ฅ and cos2 ๐‘ฅ = 1 + cos 2๐‘ฅ
2
2
1
or this identity: sin ๐‘ฅ cos ๐‘ฅ = sin 2๐‘ฅ
2
๐‘‘
๐ฎ ๐‘ก +๐ฏ ๐‘ก
๐‘‘๐‘ก
sec 2 ๐‘ฅ ๐‘‘๐‘ฅ
Then substitute: ๐‘ข = tan ๐‘ฅ
If ๐‘š is odd
= เถฑ 1 − cos2 ๐‘ฅ
๐‘˜−1
๐‘˜−1 sec 2 ๐‘ฅ ๐‘‘๐‘ฅ
= ๐ฎ′ ๐‘ก + ๐ฏ ′ ๐‘ก
= ๐‘๐ฎ′(๐‘ก)
= ๐‘“ ′ ๐‘ก ๐ฎ ๐‘ก + ๐‘“ ๐‘ก ๐ฎ′(๐‘ก)
= ๐ฎ′ ๐‘ก โˆ™ ๐ฏ ๐‘ก + ๐ฎ(๐‘ก) โˆ™ ๐ฏ′(๐‘ก)
Definite Integral of a Vector Function
๐‘
เถฑ ๐ซ(๐‘ก) ๐‘‘๐‘ก
= ๐ฎ′ ๐‘ก × ๐ฏ ๐‘ก + ๐ฎ(๐‘ก) × ๐ฏ′(๐‘ก)
= ๐‘“ ′ ๐‘ก ๐ฎ′ ๐‘“ ๐‘ก ,
๐‘Ž
๐‘
๐‘
๐‘
= เถฑ ๐‘“(๐‘ก) ๐‘‘๐‘ก ๐ข + เถฑ ๐‘”(๐‘ก) ๐‘‘๐‘ก ๐ฃ + เถฑ โ„Ž(๐‘ก) ๐‘‘๐‘ก ๐ค
(chain rule)
๐‘Ž
๐‘Ž
๐‘Ž
@SmartGirlStudy | www.SmartGirlStudy.com
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