Taller Calculo II

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    1

    1

    x3

    dx

    11

    2x2 3x3 dx

    10

    2

    2

    3 x 233

    dx

    21

    x+ 1

    x x+ 1 dx

    10

    dxx2 + 7

    4

    3

    dx

    x2 4

    10

    dx8 x2

    ln 31

    3xex dx

    0

    sen x dx

    0

    cos x dx

    4

    0

    tan x dx

    11

    (nx)1n

    n dx n N

    21

    (xm xn)3x

    dx n, m N

    21

    (xm xn)2

    x

    dx n, m N

    2c bm

    c bm

    x

    mx+b dx; m = 0, c > 0

    2c bm

    c b

    m

    ax+d

    mx+b dx; m = 0, c > 0

    10

    x

    (x+ 1)2

    dx

    1

    20

    b1 x dx

    a

    b0

    a bx dx; a,b >0

    e1

    x+ ln x

    x dx

    10

    x2

    x2 + 2 dx

    1

    20

    x3

    2 x2 dx

    10

    x

    2x2 + 3 dx

    a

    20

    xa4 x4 dx; a > 0

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    3

    2

    x2x6

    1dx

    1

    0

    arctanx2

    4 +x2 dx

    x arctan2x

    1 + 4x2 dx

    aemx dx

    ex ex dx

    x7x

    2

    dx

    e

    1

    x

    x2 dx

    ex

    ex 1 dx ex

    a bex dx dx

    2x + 3

    ax

    1 +a2x dx

    sen(a+bx)dx

    sen log x

    x dx

    sen2 x dx

    xdx

    cos2 x2

    tanx dx

    tan

    x

    x dx

    sen3 6x cos6x dx

    sen3x

    3 + cos 3x dx

    tan x

    cos2 x dx

    1 + sen 3x

    cos2 3x dx

    (2senh5x 3cosh5x) dx

    dx

    senh x cosh x

    x

    3

    1x4 4x+ 1 dx x3

    x8 + 5 dx

    dx

    ex

    1 sen x

    x+ cos x dx

    dx

    x ln2 x

    x dx

    1 x4 x2

    x2

    2

    dx

    dxex + 1

    e2x

    1 +ex dx

    sen3 x

    cos x dx

    dx

    (a+b)(a b)x2 0< b < a sen x cos x

    cos2 x sen2 x dx

    1 + 3 cos

    2

    x sen2x dx

    dx

    (1 +x2) ln

    x+

    1 +x2

    earctan x +x ln(1 +x2) + 1

    1 +x2 dx

    x2

    1 x2 dx

    x32 x2 dx

    dx

    x

    x2 1

    dx

    x2

    4 x2

    x2

    a2

    x dx

    x2 + 1

    x dx

    1 x2 dx

    25

    x2

    x dx

    dx

    x2 1

    dx

    x2 2x+ 5

    x2

    + 4 dx

    x

    1 +x4 dx

    11

    x2

    25 x2 dx 1

    0

    x2

    (4 x) 32dx

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    1

    1x3

    16 x2

    dx

    1

    0

    dx

    x

    25 4x

    2

    arctan x dx

    x sen x dx

    x dx

    ex

    x2e3x dx

    x

    3

    ex

    3 dx

    ln2 x dx

    ln x

    x dx

    x arc senx dx

    x

    sen2 x dx

    ex sen x dx

    eax sen bx dx

    cos

    x dx

    xex dx

    x sec x tan x dx

    x

    2

    ln x dx

    arctan x dx

    (ex + 2x)

    2dx

    x sen x cos x dx

    x2

    + 5x+ 6

    cos2x dx

    e1

    x2 ln x dx

    0

    sen(3x)cos(5x)dx

    1

    11

    12x2 + 3653x+ 2 dx

    e1

    ln 3x dx

    0

    (2x 5)sen x dx

    10

    (2x+ 4)e2x+4 dx

    1

    1

    arctan x dx

    1

    20

    arcsen x dx

    n, m N

    senn x dx= senn1 x cos x+ (n 1)

    senn2 x cos2 x dx

    senn x dx= sen

    n1 x cos x

    n +

    n 1n

    senn2 x dx

    cosn x dx= cos

    n1

    x sen xn

    + n 1n

    cosn2 x dx

    senn+1 x

    cosm+1 x dx=

    1

    m

    senn x

    cosm x n

    m

    senn1 x

    cosm1 x dx

    cosm+1 x

    senn+1 x dx=

    1

    n

    cosm x

    senn x m

    n

    cosm1 x

    senn1 x dx

    xn ln x dx

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    2

    1

    x3ex2

    dx

    21

    ln2 x

    x2 dx

    10

    sen2 x

    ex dx

    2

    1

    x2

    (x2 + 1)2 dx

    21

    x29 x2 dx

    40

    x2 arctan 3x dx

    2

    0

    (arc senx)2 dx

    a

    20

    a2 x2 dx

    4

    8

    arcsen

    x1 x dx

    2

    1x

    2 + 2x+ 3 ln x dx

    10

    x(arctan x)2 dx

    a0

    dx

    (a2 +x2)2

    b

    h

    A= 12

    bh

    r >0 A= r2

    B

    b

    h

    A= 12

    (B+b)h

    r V = 43

    r3

    r h V = 1

    3r2h

    (x, y)

    x2 y2 36 0, y2 9x.

    a >0 y = a sen x

    y= cos x, 0 x 2 .

    x2

    9 +

    y2

    4 = 1,

    x2

    4 +

    y2

    9 = 1.

    y = x2 x = y2 x= 1

    y = 6 3x x2 x+y 3 = 0

    x= 3

    y= 0

    x2 +y2 = 9 x= 4

    h 2a 2b

    x x 2x2 x= 0 x= a

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    y= 3 2x x2 (2,5) y

    y2

    =x(x 1)2

    2

    6 x= y2

    y = cos x x y= sen x x=

    2

    a2y2 =ax3 x4 x (a > 0) x y= a cosh xa (a > 0)

    y = sen x(0 x 2

    ) y= 1 y

    x= 4 y2

    y [2, 2] x

    2a a a > 0 h

    y

    x = y2 x = 8 y2

    a > 0

    y = sen x

    y= 0

    x= 0

    x= 1/4

    x

    y

    y2 = 4ax a >0 x= a

    x2

    4 +y2 = 1 x= 3

    f(x) = 3x3 x2 10x g(x) = 2x x2

    x

    y2 = x

    y2 = 2x x2 x= 3

    f(x) = 1 x/2 g(x) = 1 +x/2 x= 0 x x= 2

    y= x3 +x+ 1 y= 1 x= 1

    f(x) =x3

    6 +

    1

    2x

    1

    2, 2

    a > 0 b > 0

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    y= x3 y= x2

    y= x+ 6

    y =

    log x

    y= x2 +x+ 3 y= x3 + 3x2 2x+ 3 (1, e) (e, 1) y= ex y= log x y= 1 x

    x2 +y2 36 y2 9x

    r h

    r h

    4

    y = 2

    y = x2

    + 2x

    x2 +y2 +z2 1 x2 +y2 r2 r (0, 1)

    [0, 4] f(x) = 42

    3 x3/2 1

    y= 3x (1, 3) (2, 6)

    9y2 = 4x3

    8y = x4 + 2x2 x = 1 x= 2

    y = 13

    (x2 + 2)3/2 x = 0

    x= 3

    x2/3 +y2/3 = 1 x= 1/8 x= 1

    10

    dxx

    21

    dx

    x

    10

    dx

    xp

    30

    dx

    (x 1)2

    10

    dx1 x2

    1

    20

    dx

    x ln x

    2

    0

    cot x

    10

    dx

    x3 5x2

    0

    dx

    x

    0

    dx

    xp

    0

    sen x

    a

    dx

    x ln x (a > 1)

    a

    dx

    x ln2 x (a > 1)

    0

    ekx dx

    0

    arctan x

    x2 + 1 dx

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    2

    dx

    (x2

    1)2

    0

    dx

    x3 + 1

    dx

    1 +x2

    dx

    x2 + 4x+ 9

    c

    2

    cx

    x2 + 1 1

    2x+ 1

    dx

    c

    c

    1

    x

    2x2 + 2c c

    x+ 1

    dx

    c

    c

    0

    11 + 2x2

    cx+ 1

    dx

    c

    1

    (x+a)(x+b)

    x2 5x+ 9x2 5x+ 5

    1

    (x 1)(x+ 2)(x+ 3)

    2x2 + 41x 91(x 1)(x+ 2)(x 4)

    5x3

    + 2x3 5x2 + 4x

    1

    x(x+ 1)2

    x3 114x3 x

    x2 8x+ 7(x2 3x 10)2

    x3 +x+ 1

    x(x2 + 1)

    x4

    x4 1

    1

    (x2 4x+ 3)(x2 + 4x+ 5)

    1x3 + 1

    x3 + 1

    (x2 4x+ 5)2

    1

    (x+ 1)(x2 +x+ 1)2

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    cos3 x dx

    sen5 x dx

    sen2 x cos3 x dx

    sen3

    x

    2cos5

    x

    2 dx

    cos5 x

    sen3 x dx

    sen4 x dx

    1

    sen4

    x

    dx

    1

    sen2x cos4 x dx

    sec5 4x dx

    tan2 5x dx

    cot3 x dx

    cot4 x dx

    tan3

    x

    3+ tan4

    x

    4

    dx

    x sen2 x2 dx

    sen5 x 3

    cos x dx

    sen3x cos5x dx

    sen10x sen15x dx

    cos

    x

    2cos

    x

    3 dx

    sen

    x

    3cos

    2x

    3dx

    cos(ax+b) cos(ax b)dx

    sen ax sen(ax+b)dx

    an= 1

    log(n+k) bn=

    1

    log n k R+

    an= 1

    log(kn) bn=

    1

    log n k R+ k = 1

    n

    p

    k

    an= p(n) bn= p(n+k)

    an= apnp + ap1n

    p1 + ap2np2 + + a1n + a0 bn= apnp ap= 0

    an= 1

    (n+ 1) n bn= 1

    n1

    = 0

    n=1

    n

    7n+ 4

    4n2

    n=2

    1

    (ln n)lnn

    n=2

    ln

    n+ 1

    n 1

    n

    n=1

    n

    (n+ 1)3

    n=1

    n+ 2(n+ 1)

    n+ 3

    n=1

    4n2 n+ 3n3 + 2n

    n=1

    2n1

    (3n+ 2) n4/3

    n=1

    3n

    n5n

    n=1

    n+

    n

    2n3 1n=2

    1

    ln n

    n=1

    ln na

    n!

    n=1

    ln n

    2n3 1n=1

    (n!)24n

    (2n)!

    n=1

    lnn+ 1

    n+ 2

    n=1

    an

    n a > 0 a = 1

    n=1

    n!an

    nn a >0 a = 1

    n=1

    na

    n!

    n=1

    sen4 n

    n2

    n=1

    nj=1

    (a+j)

    n!

    n=1

    nan

    a > 0 a

    = 1

    n=1

    (ln n)n

    n=1

    n+ 1 n

    1

    2 3+ 2

    3 4+ 2

    5+

    22 + 1

    2 52 +32 + 1

    3 53

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    1 + 2!

    22+

    3!

    33+

    4!

    44

    1

    2+

    2

    3+

    3

    4

    n=1

    nn

    n!

    n=1

    n2en

    n=2

    1

    (n log n)p

    n=1

    1

    (n+ 1)(n+ 2)

    n=1

    1

    2n(n+ 1)

    n=0

    12

    100n

    n=0

    3n + 4n

    5n

    n=2

    ln

    n+ 1

    n 1

    n=0

    n

    2n1+

    3

    2n

    n=1

    ln(n+ 1)2

    n(n+ 2)

    n=0

    1 2n3n

    n=1

    n

    2n 1+ 3

    2n

    n=0

    2n +n(n+ 1)

    2n+1n(n+ 1)

    n=0

    ln

    1 +

    1

    n

    n

    (n+ 1)

    [ln(n+ 1)n+1] ln nn

    n=1

    1

    n 1

    n!

    n=1

    n3

    2n

    n=1

    (1)n ln nn

    n=1

    n!

    (2)n

    n=1

    (1)nn+

    n+ 1

    n=1

    sen

    4n2

    n=1

    (1)nn(n+ 1)

    n=1

    (1)n n2n

    n=1

    (1)n 1n

    1 13

    +1

    5 1

    7+

    1

    2 l n 2 1

    3 l n 3+

    1

    4 l n 4 1

    ln 5+

    1

    4 1

    6+

    1

    8 1

    10+

    n=1

    1

    n2nxn

    n=1

    1

    nxn

    n=1

    (1)nn

    x2n

    n=1

    1

    n2xn

    n=1

    (n!)2

    (2n)!x2n

    n=0

    n2 + 1

    (n+ 1)!xn

    n=0

    2n

    n+ 1xn

    n=2

    nxn

    n=2

    1

    n2n(x 3)n

    n=2

    2n

    n+ 1

    x+

    1

    2

    n

    n=1

    xn ln n

    n=1

    n2

    2nxn

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    n=0

    2n 13n+ 1

    xn

    n=1

    nn

    n!

    x+

    1

    2

    n

    n=2

    2 5 8 11 (3n 1)3 7 11 15 (4n 1) x

    n

    n=0

    (1)nn!n2

    (x 1)n

    n=0

    (1)nn+ 1

    (x 1)n

    n=0

    (1)nn33n

    (x+ 3)n

    n=0

    xnn

    ln

    2n+ 1

    n

    x

    f(x) = ex

    f(x) = sen x

    f(x) = ln(1 x) f(x) =

    1

    1 x

    f(x) = 1

    1 +x2

    f(x) = arctan x

    x

    f(x) = 1

    1 + sen x f(x) =

    cos x

    1 + sen x f(x) =

    cos x

    1 2x