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Title: Summary of differentiation and integration techniques (Calculus)
Description: This document encompasses the techniques required for differentiation and integration in the A Level syllabus.
Description: This document encompasses the techniques required for differentiation and integration in the A Level syllabus.
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Techniques of Differentiation
(
First Principles:
)
( )
Chain Rule:
(
Product Rule:
Quotient Rule:
)
( )
Differentiation of Standard Function
( )
[ ( )]
[ ( )]
( )
Trigonometric Functions
[
( )]
[
( )]
[
( )]
( )
( )
( )
( )
[
( )
[
( )]
[
( )
( )
( )]
( )]
( )
( )
( )
( )
( )
( )
( )
Exponential and Logarithmic Functions
[
[
( )
( )
]
( )
( )
( )
( )]
[
[
( )
( )
]
(
) ( )
( )
( )
( )]
Inverse Trigonometric Functions
[
]
[
]
[
]
[
√
√
( )]
[
( )]
[
( )]
Parametric Differentiation
( )
√
√
( )
[ ( )]
( )
[ ( )]
( )
[ ( )]
( )
( )
Techniques of Integrations
Basic Integrals
∫
| |
∫
∫
∫
Integration by Standard Forms
[ ( )]
∫ ( )[ ( )]
∫ ( )
( )
∫
( )
( )
( )
| ( )|
( )
∫ ( )
( )
Trigonometric Integral
*MF 15 is without the | | sign
∫ ( )[
( )]
∫ ( )[
( )
( )]
( )
( )
∫ ( )[
( )
( )]
∫ ( )[
∫ ( )[
∫ ( )[
( )]
( )
( )
( )]
( )
( )]
( )
Integrals of tan x, cot x, cosec x and sec x can be found in MF15
Using Factor Formula
*take note that (
(
∫
∫
)
(
)
∫[
)
(
(
)
)
(
(
)
) ]
(
)
in MF15
**refer to MF15 for the others**
Standard Integrals
An example only
...
( )
When using MF15, ensure coefficient of
2
[f(x)] is ±1
( )
[ ( )]
Rational Functions of the Form ∫
Let f(x) = A[g’(x)] + B
Compare to find A and B
( )
( )
2
, where g(x) = ax + bx + c
Ans:
| ( )|
∫
( )
complete the square for g(x) and make to
standard integral
...
Differentiate the substitution to replace dx with du
Integration by Parts
Formula: ∫
**Use the LIATE rule for the choice of u
e
...
∫
∫(
)( )
Techniques of Differentiation
(
First Principles:
)
( )
Chain Rule:
(
Product Rule:
Quotient Rule:
)
( )
Differentiation of Standard Function
( )
[ ( )]
[ ( )]
( )
Trigonometric Functions
[
( )]
[
( )]
[
( )]
( )
( )
( )
( )
[
( )
[
( )]
[
( )
( )
( )]
( )]
( )
( )
( )
( )
( )
( )
( )
Exponential and Logarithmic Functions
[
[
( )
( )
]
( )
( )
( )
( )]
[
[
( )
( )
]
(
) ( )
( )
( )
( )]
Inverse Trigonometric Functions
[
]
[
]
[
]
[
√
√
( )]
[
( )]
[
( )]
Parametric Differentiation
( )
√
√
( )
[ ( )]
( )
[ ( )]
( )
[ ( )]
( )
( )
Techniques of Integrations
Basic Integrals
∫
| |
∫
∫
∫
Integration by Standard Forms
[ ( )]
∫ ( )[ ( )]
∫ ( )
( )
∫
( )
( )
( )
| ( )|
( )
∫ ( )
( )
Trigonometric Integral
*MF 15 is without the | | sign
∫ ( )[
( )]
∫ ( )[
( )
( )]
( )
( )
∫ ( )[
( )
( )]
∫ ( )[
∫ ( )[
∫ ( )[
( )]
( )
( )
( )]
( )
( )]
( )
Integrals of tan x, cot x, cosec x and sec x can be found in MF15
Using Factor Formula
*take note that (
(
∫
∫
)
(
)
∫[
)
(
(
)
)
(
(
)
) ]
(
)
in MF15
**refer to MF15 for the others**
Standard Integrals
An example only
...
( )
When using MF15, ensure coefficient of
2
[f(x)] is ±1
( )
[ ( )]
Rational Functions of the Form ∫
Let f(x) = A[g’(x)] + B
Compare to find A and B
( )
( )
2
, where g(x) = ax + bx + c
Ans:
| ( )|
∫
( )
complete the square for g(x) and make to
standard integral
...
Differentiate the substitution to replace dx with du
Integration by Parts
Formula: ∫
**Use the LIATE rule for the choice of u
e
...
∫
∫(
)( )
Title: Summary of differentiation and integration techniques (Calculus)
Description: This document encompasses the techniques required for differentiation and integration in the A Level syllabus.
Description: This document encompasses the techniques required for differentiation and integration in the A Level syllabus.