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    Ejercicios Resueltos Cálculo Diferencial e Integral

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    Ana Sofia Cruz Romero
    This document contains 15 problems involving calculating derivatives and integrals using differentiation and integration rules. Some examples include finding the derivative of functions using the product, quotient and chain rules as well as taking integrals of rational functions and trigonometric functions. The problems progress from simpler applications of basic rules to more complex expressions requiring multiple steps to derive the solution.

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    Ejercicios Resueltos Cálculo Diferencial e Integral

    Uploaded by

    Ana Sofia Cruz Romero
    64 views4 pages
    This document contains 15 problems involving calculating derivatives and integrals using differentiation and integration rules. Some examples include finding the derivative of functions using the product, quotient and chain rules as well as taking integrals of rational functions and trigonometric functions. The problems progress from simpler applications of basic rules to more complex expressions requiring multiple steps to derive the solution.

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    Original Title

    EJERCICIOS RESUELTOS CÁLCULO DIFERENCIAL E INTEGRAL

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    This document contains 15 problems involving calculating derivatives and integrals using differentiation and integration rules. Some examples include finding the derivative of functions using the product, quotient and chain rules as well as taking integrals of rational functions and trigonometric functions. The problems progress from simpler applications of basic rules to more complex expressions requiring multiple steps to derive the solution.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    64 views4 pages

    Ejercicios Resueltos Cálculo Diferencial e Integral

    Uploaded by

    Ana Sofia Cruz Romero
    This document contains 15 problems involving calculating derivatives and integrals using differentiation and integration rules. Some examples include finding the derivative of functions using the product, quotient and chain rules as well as taking integrals of rational functions and trigonometric functions. The problems progress from simpler applications of basic rules to more complex expressions requiring multiple steps to derive the solution.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    of 4

    LABORATORIO #2 CÁLCULO DIFERENCIAL E INTEGRAL

    1 ¿ f ( x )=( x−senx )( x +cosx ) Regla del producto

    f ´ ( x )=[ x−sen ( x ) ] [ 1+ ( −sen ( x ) ∙ 1 ) ] + [ x +cos ( x ) ] [1−( cos ( x ) ∙1 ) ]


    f ´ ( x )=[ x−sen ( x ) ][ 1−sen ( x ) ] + [ x+ cos ( x ) ] [1−cos ( x )]
    2 2
    f ´ ( x )=x−xsen ( x ) −sen ( x ) + sen ( x ) + x−xcos ( x ) +cos ( x )−cos ⁡( x )
    2 2
    f ´ ( x )=2 x−xsen ( x )−sen ( x )+ sen ( x ) −xcos ( x ) +cos ( x )−cos ⁡( x )

    −1
    2 ¿ f ( x )=3 sen ( x ) Regla del múltiplo constante
    u=x
    du=1

    (√ )
    1
    f ´ ( x )=( 3 ) 2
    1− ( x )
    3
    f ´ ( x )=
    √ 1−( x ) 2

    2 sen ( 3 x )
    3 ¿ f ( x )=e
    2 sen ( 3 x )
    f ´ ( x )=e ∙ ( 2 ) (cos ( 3 x ) ∙ 3)
    2 sen ( 3 x )
    f ´ ( x )=e ∙ 6 cos ( 3 x )
    2 sen ( 3 x )
    f ´ ( x )=6 e cos ( 3 x )

    cotx
    4 ¿ f ( x )= Regla del cociente
    1−senx

    f ´ ( x )=(1−sen x)(−csc 2 x ∙ 1)−( cot x )¿¿


    f ´ ( x )=(1−sen x)(−csc 2 x )−(cot x )¿ ¿
    −csc 2 x+ sen x csc2 x +cot x cos x
    f ´ ( x )= 2
    (1−sen x )
    2 2
    −csc ( x )+ sen ( x ) csc ( x ) +cot x cos x
    f ´ ( x )= (revisar)
    (1−sen x )2

    2 cscx−1
    5 ¿ f ( x )= Regla del cociente
    cscx+2
    ( csc x+ 2 )( 2 ∙−csc x cot x ∙1 ) −(2 csc x−1)(−csc x cot x ∙ 1)
    f ´ ( x )= 2
    (csc x+ 2)
    ( csc x+ 2 )(−2 csc x cot x )−(2 csc x−1)(−csc x cot x )
    f ´ ( x )=
    (csc x +2)2
    ( ) −2 csc 2 x cot x−4 csc x cot x+ 2 csc 2 x cot x−csc x cot x
    f´ x =
    (csc x+ 2)2
    −5 csc x cot x
    f ´ ( x )= 2
    (csc x +2)

    6 ¿ f ( x )=cotx cotx Regla de la cadena

    f ( x )=(cot x)2
    f ´ ( x )=2 ∙(cot x) ∙ (−csc 2 x ∙ 1 )
    f ´ ( x )=−2cot x csc 2 x

    7 ¿ f ( x )=cos (lnx )
    1
    f ´ ( x )=−sen (lnx) ∙
    x
    −sen(lnx)
    f ´ ( x )=
    x

    8 ¿ f ( x )= √ tan ( 4 x )

    cosx−1
    9 ¿ f ( x )=
    senx

    [ ]
    2 3
    x ( 2 x−1 )
    10 ¿ f ( x )=ln
    ( x +5 )2

    11 ¿ f ( x )=ln 3
    √ x+1
    2
    x +1

    12 ¿ f ( x )=x 2 e x −2 x e x + 2 e x
    x−tanx
    13 ¿ f ( x )=
    x−senx

    3x −2 x
    e −5 e
    14 ¿ f ( x )=
    e4 x

    x 2 +2 x
    15 ¿∫ dx
    √ x 3+ 3 x 2 +1
    dU
    U =x3 +3 x 2+ 1dU =3 x 2+6 x dxdU =3 ( x 2+ 2 x ) dx =x 2+ 2 x dx
    3
    dU 1 1
    1
    1 2 1 2
    3 1 dU 1 1 U 1 2U 3 2 2
    ∫ 1
    = ∫ 1 = ∫ dU ∙U = ∙
    3 3 3 1
    +C= ∙ 2U +C=
    3 3
    2
    +C ¿ 2(x +3 x +1) + C 2

    U2 U2 3
    2

    2
    x
    16 ¿∫ 4 dx
    √ x 3 +2
    dU 3 3
    −1 4 4
    dU 3 1 dU 1 1 U 4U
    =x dx∫
    3 ∫ 14 3 ∫
    2
    U =x3 +2dU =3 x 2 dx = = U 4
    dU = ∙ +C= +C
    3 1
    4
    3 3 9
    (U ) U 4

    17 ¿∫ √ 1+ tan 2 ( 6 x ) dx

    ( 1+lnx )3
    18 ¿∫ dx
    x

    ( )( )
    3 2
    1 x −1
    19 ¿∫ x + 2
    dx
    x x2
    2
    1 1 1 x −1 3
    U =x+ dU =1−x 2 dxdU = − 2 dx dU = 2 dx ∫ U 2 dU
    x 1 x x
    5 5
    U2 2U 2 2 1 5
    +C= +C= (x + ) 2 +C
    5 5 5 x
    2

    cot ( lnx ) dx
    20 ¿∫
    x

    csc 2 ( 3 x )
    21 ¿∫ 3
    dx
    cot (3 x )
    −dU 2
    U =cot ( 3 x )dU =−csc 2 ( 3 x ) ( 3 ) dx =csc ( 3 x ) dx
    3
    −dU
    2
    csc ( 3 x ) −2 −2 1
    3 −1 −3 −1 U +U 1 ¿ +C
    ∫ 3
    =∫ 3
    =
    3 ∫ dU ∙ U = ∙
    3 −2
    + C=
    6
    +C= 2
    +C 6[cot (3 x)]
    2

    [ cot ( 3 x ) ] U 6U

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