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    Formulas

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    يوسف العامري
    This document contains formulas for differentiation and integration. It lists 15 common differentiation formulas including formulas for exponential, logarithmic, trigonometric and inverse trigonometric functions. It also provides 15 integration formulas that are the inverse of the differentiation formulas. Additionally, it supplies the product rule, quotient rule and chain rule for differentiation along with other trigonometric relations and values in 3 tables.

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    © All Rights Reserved
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    Formulas

    Uploaded by

    يوسف العامري
    561 views3 pages
    This document contains formulas for differentiation and integration. It lists 15 common differentiation formulas including formulas for exponential, logarithmic, trigonometric and inverse trigonometric functions. It also provides 15 integration formulas that are the inverse of the differentiation formulas. Additionally, it supplies the product rule, quotient rule and chain rule for differentiation along with other trigonometric relations and values in 3 tables.

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    This document contains formulas for differentiation and integration. It lists 15 common differentiation formulas including formulas for exponential, logarithmic, trigonometric and inverse trigonometric functions. It also provides 15 integration formulas that are the inverse of the differentiation formulas. Additionally, it supplies the product rule, quotient rule and chain rule for differentiation along with other trigonometric relations and values in 3 tables.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    561 views3 pages

    Formulas

    Uploaded by

    يوسف العامري
    This document contains formulas for differentiation and integration. It lists 15 common differentiation formulas including formulas for exponential, logarithmic, trigonometric and inverse trigonometric functions. It also provides 15 integration formulas that are the inverse of the differentiation formulas. Additionally, it supplies the product rule, quotient rule and chain rule for differentiation along with other trigonometric relations and values in 3 tables.

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    Calculus Formulae

    DIFFERNTIAL FORMULAS: INTEGRATION FORMULAS:


    d x
    n+1

    1. dx
    (ax )=a ∫ x dx= n+1
    n
    +C
    1.
    d n eax
    ( x )=nx n−1
    2. dx ∫ e dx= a +C
    ax
    2.
    d x
    (a )=a x ln a ax
    3. dx ∫ a dx =ln a +C
    x
    3.
    d ax
    (e )=aeax −cos ax
    4. dx ∫ sin axdx = a +C
    4.
    d
    (sin ax )=a cosax sin ax
    5. dx ∫ cos axdx= a +C
    5.
    d
    6. dx
    (cos ax )=−asin ax
    6. ∫ tan xdx=ln|sec x|+C
    d
    ( tan ax )=a sec 2 ax 7. ∫ cot xdx=−ln|csc x|+C
    7. dx
    d
    ( tan x )=sec 2 x 8. ∫ sec xdx=ln|sec x+tan x|+C
    8. dx
    d 9. ∫ csc xdx=−ln|csc x+cot x|+C
    (cot x )=−csc 2 x
    9. dx 10. ∫ sec2 xdx=tan x+C
    d
    (sec x )=sec x tan x 11. ∫ csc2 xdx=−cot x+C
    10. dx
    d
    (csc x )=−csc x cot x 12. ∫ sec x tan xdx=sec x+C
    11. dx ∫ csc xcot xdx=−csc x+C
    d 1 13.
    ( ln x )= 1
    12. dx x ∫ x+ a dx =ln|x+ a|+C
    (Note- 14.

    d
    log e x=ln x )
    15.
    ∫ 2
    1
    x +a
    2
    1
    dx= tan−1
    a
    x
    a
    +C ()
    ( sin−1 x )= 1 2
    13. dx √ 1−x
    d
    ( tan−1 x ) = 1 2 ∫ 1 2 2 dx =sin−1( ax )+C
    14. dx 1+ x √ a −x
    d
    ( sec−1 x ) = 12 16.
    or =cos−1 ( xa )+C
    15.
    dx |x| √ x −1
    1
    ∫ dx=sec−1 ( x )+C
    17. |x|√ x 2−1

    1|Page
    1
    ∫ dx=ln|x + √ x 2 +a2|+c
    18. √x +a2 2

    Differential of product of two functions (Product Rule):


    d d d
    [u( x ). v( x )]= [u ( x )] . v ( x )+u( x ). [ v( x)]
    dx dx dx

    In short [ u. v ] '=[ u ] ' .v+u. [ v ] '


    Differential of Quotient of two functions (Quotient Rule):

    dx v ( x ) [ ]
    d u( x ) v ( x )u '( x )−u ( x ). v '( x )
    =
    [ v (x )]
    2

    Chain Rule of differentiation


    d d
    [ f ( g( x ))]=f ' (g ( x )). (g (x ))
    dx dx
    Reverse Chain Rule of Integration
    ∫ f ' ( g( x)).g ' ( x)dx= f ( g( x))+C
    Table 2: Other Useful Relations and Results
    2 2 2
    1. sin 2 A=2 sin A cos A 12. ( A±B ) = A +B ±2 AB
    2 2 2 2
    2. sin x +cos x=1 13. A −B =( A +B)( A−B)
    1−cos 2 x ln A+ln B=ln A . B
    sin2 x= 14.
    3. 2 ln Ak =k ln A
    15.
    1+cos 2 x
    cos 2 x = ln A−ln B=ln
    A
    4. 2 B
    16.
    2 2
    5. sec x=1+tan x ln e = A
    A
    2 2
    17.
    csc x=1+cot x
    6.
    18. e k ln A= A k
    1 log e A
    sin x= ln A
    7. csc x log B A= =
    19.
    log e B ln B
    1
    cos x= a c d abe±ce±db
    2

    8. sec x ± 2± =
    20. b b e b2 e
    sin x 22
    tan x= π= =3. 14 28
    9. cos x 21. 7
    0
    22. a =1 where a≠0

    2|Page
    cos x 23. ln 1=0
    cot x=
    10. sin x
    1
    tan x=
    11. cot x
    sin ( A+ B ) +sin ( A−B )
    sin A cos B=
    24. 2
    sin ( A+B )−sin ( A−B )
    cos A sin B=
    25. 2
    cos ( A + B ) +cos ( A−B )
    cos A cos B=
    26. 2
    cos ( A−B )−cos ( A+ B )
    sin A sin B=
    27. 2
    n
    28. sin nπ =0 , cos nπ =(−1 )

    Table 3: Trigonometric Values

    0 π /6 π /4 π /3 π /2 π nπ

    sin 0 1/2 1/ √2 √3 1 0 0
    2
    cos 1 √ 3/2 1/ √2 1/2 0 -1 (-1)n
    tan 0 1/ √3 1 √3 ∞ 0 0

    3|Page

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