Analisi Matematica 3 Esercizi

= x⁺³ ∑_∞⁰ (2n+m) / m+1

= 2n / m+1 e^inx e ^{-y ex+3∑_xt3/m+1}

e ^x+3 ∑_∞⁰ y ^m / y^m+1

= e ^-ln(1-ex)

e ^x+3 (-ln(1-y))

Serie di Fourier

f(x) = x³

z₀ = 1/π ^π∫ x³ dx - 1/π ∫_π/-π [ x⁺⁴ / π ]^π_-π = 0

c_n = ∫_-π^π x³ cos (nx) dx = ∫ x³ sin (nx) / n - ∫ 3x x cos (nx) / n dx + 1/π =

-1/π [ x³ sin (nx) / n - 3x² sin (nx) / n² - ∫ -π_π 6x sin (nx) / n² dx ] =

= -1/π [ x³ sin (nx) / n - 3x² sin (nx) / n² + ∫_-π^π 6x sin (nx) / n² dx ] =

= -1/π [ x³ sin (nx) / n - 3x² sin (nx) / n² + (6x sin (nx) / n³) - ∫_-π^π 6 cos (nx) / n³ dx ] =

= -1/π [ x³ sin (nx) / n - 3x² sin (nx) / n² - 6x cos (nx) / n³ + ∫_-π^π 6 cos (nx) / n³ dx ]

Serie di Fourier

f(x) = x³

z₀ = ¹/_π ∫_-π^π x³ dx = [^π/_π(^x4/₄)]_-π^π = [^π4/₄ - ^π4/₄] = 0

c_n = ¹/_π ∫_-π^π x³ cos(nx) dx = [x³ sin(nx)/n]_-π^π - 3x² ∫_-π^π cos(nx)/n dx

= [^-1/_π { [x³ sin(nx)/n - 3x² sin(nx)/n²}- ∫_-π^π 6x sin(nx)/n² dx]]

= [^-1/_π { [x³ sin(nx)/n - 3x² sin(nx)/n²}-∫_-π^π 6x sin(nx)/n² dx]

= [^-1/_π [x³ sin(nx)/n - 3x² sin(nx)/n² + (6x cos(nx)/n³]_-π^π + ∫_-π^π 6 cos(nx)/n³ dx]

= [^-1/_π [x³ sin(nx)/n - 3x² sin(nx)/n² - 6x cos(nx)/n³ +∫_-π^{π 6 cos(nx)/n3 dx]}

= 0

ϕ(t) = (t, cos(t), cos₂(t))

Dϕ(t) = (1, -sin(t), -2 sin(t) )

∫₀^2π ||Dϕ(t)|| dϕ = ∫₀^2π √1+(-sin(t))² + (-2 sin(t) )² =

= ∫₀^2π √1+ sin²(t) + 4 sin²(t) = ∫₀^2π√1+1-cos²(t)+4-4cos²(t)

Curva

γ(t) = (5cos(t), 5sin(t), -sin(5t)) con 0 < t < 2π

Dγ(t) = (-5sin(t), 5cos(t), -5cos(5t))

|Dγ(t)| = ∫₀^2π √25sin²(t) + 25cos²(t) + 25cos²(5t) dt =

= ∫₀^2π √25 - 25cos²(t) + 25cos²(t) + 25cos²(5t) = ∫₀^2π √25 + 25cos²(5t) ≈ 38,2

∫_τ ln (3x + y) dx dy

OA: y - 2 = x - 1 -> y = 2/3x - 2x = 0

2x - y = 0

OB: y - 1 = x - 2 -> -2y + 2x - x = 0

x - 2y = 0

∆ x = 4 - 1 = 3

1 - 4 = -3 -> -x + 2y -> 2x + 3y = 3

{u = 2x - y

v = -x + 2y

2x = u + y

y = (u + 2v)

-v + 2y = u + y

y = (u + 2v)/3

J = |1 2/3|

|2/3 1/3|

= 1/3

∫³₂ ∫^3-v₀ ln ((6u + 3v + u + 2v)/3) du dv -

∫³₀ ∫ln (7u + 5v) dx - ln (3) du dv

l'(x) = ln (7u + 5v)

g'(x) = 7

g(x) = 7u + 5v

∫(7u + 5v) ln (7u + 5v) - ∫(7u + 5v)/(7u + 5v) -

[∫(7u + 5v) ln (7u + 5v)]³₀ - 1

[ (7u + 5v) ln(7u + 5v) ]^7-v - [U]^7-v

[ (7u + 5v) ln(7u + 5v) - 5v ln(5v) ]₃

[ (7 - 7v + 5v) ln(7 - 7v + 5v) ] - 5v ln(5v) ] - 3 + v

[ (7 - 2v) ln(7 - 2v) - 5v ln(5v) ] - 3 + v - 3 ln(3) + v ln(3)

g(x) ln(7 - 2v) g'(x) = x - 2 - 2v

f(x) ln(x) x^-1 - 2x f(x) = ln(x) g'(x) = x

f'(x) = ¹⁄_x g⁸(x) = ^x⁄₂

¹⁄₁₄ ∫₁²⁷ x ln(x)

[ x ln(x) ]₁⁷⁵ x dx

1/14 [ x ln(x) ]₂₁²⁷ x dx

[ x ln(x) ]₂₇²¹ x dx

[ 27 ln(27) - 15 ln 75 ] - 27 x 75

¹⁄₁₄ ln(27) - 15 ln(75) - 27 + 5 = ¹⁄₂₄

¹⁄₂₈ [ x² ln(x) ]₁²² x dx

1/28 [ 441 ln(27) - 725 ln(75) ] - ⁷⁴⁷¹⁄₂ ²⁸⁵⁵⁄₂ x 2 x 2

3/6 5v ln(5v) 5v = x

f(y) = ln(x) g'(x) = x

f'(x) = ¹⁄_x g'(x) = ^x⁄₂

∫₀⁷⁵ x² ln(x) ∫₀⁷⁵ x dx

-₁/₇₀ [225 ln (15)] - [^x/₂] ¹⁵₀ = -₁/₇₀[225 ln (15)] - [^{225 5}/₂]) = -7,09

-∫³₀ 3 = - 3 []³₀ = -3 . 3 = -9

∫vdv = ∫³₀ ^v2/₂ dv = [^v2/₂] ³₀ = ⁹/₂

∫₀³ 3 ln (3) dv = 3 ln (3) ∫₀³ v dv = ¹/₃[2 ln (3)

∫₀³ vln (3) = ⁹/₂ ln (3)

92 - 7,09 - 9 + ⁹/₂ - 9 ln (3) + ⁹/₂ ln (3) = -⁵⁴⁶/₃, 182

∫ ln (3x+4y) dx

T = {0, A(4, -1), B (3, 2)}

OA: y + 1 = _x/⁴

→ 4y = x - 4

→ x + 4y = 0

OB: y - 2 = _{x - 3}/³

→ 3y - 6 = 2x - 6

→ 2x - 3y = 0

x + 4y = 3 + 8 = 11 √

2x - 3y = 8 + 3 = 11 √

u = x + 4y → {x = u - 4y}

v = 2x - 3y → {x = _{v + 3y}/²}

v + 3y = 2u - 8y

4y = 20 - v

{y = 2u - v}

{y = _{2u - v}/¹¹}

(y = _{2u - v}/¹¹ 3 →) {x = _{u + 3v}/²}

y = _{2u - v}/¹¹

x = _{4v + 3u}/¹¹

J = |₂/¹¹ - ₁/¹¹| |₈/¹¹ + ₃/¹¹ - ₁/¹¹|

|₃/¹¹ 4 |

|₁₁ |

∫ ₀^11-v ∫ ₀^11-u ln (_{3u + 12 v + 2u - v}/¹¹) dudv = ₁/¹¹ ∫ ₀^11-u ln (14u + 11v) - ln (11) dudv

∫ ₀^11-u ln (14u + 11v) dv

f(x) = ln(x)

g'(x) = 14

f'(x) = ₁/^{14u + 11v}

g*(x) = 14u + 11v

∫₀^11-u [(11v + 11u)(ln (14u + 11v))]

∫₀^11-u(14u + 11v)(ln (14u + 11v)) dv - ∫₀∫_v dv

- {|(121 - 33u + 33u)(ln(321 - 33u + 11v))|}

= {*|∫₀¹¹ ∫₀ 2u ln (121) - 11 + v dv|}

∫ ln(t+1) du = -ln(t+1)∣₀^11-v u du = -ln(t+1)(11-v) = -11ln(t+1) + vln(t+1)

∫₀¹¹ 11 ln(t+1) - 11 + v + 11 ln(t+1) + v ln(t+1) du

∫₀¹¹ v du - ∫₀¹¹ v²/₂ = ∫₀¹¹ v²/₂ - ∫₀¹¹ 11²/₂

g'(x) = v

g(x) = v²/₂

∫ 11 ln(t+1) dv v ln(t+1) dv

f(x) = ln(t+1)

f'(x) = ¹/_t+1

∫_a¹¹ [v²/₂ ln(t+1)] - ∫₀¹¹ -v²/₂ dv = [11²/₂ ln(t+1)] - [-11/₆ v²/₂]¹¹₀ =

-¹¹/₆ + 11²/₂ ln(t+1) - 11/₂

∫ ln(3x+y) dx dy

OA:

OB:

x + 4y

3x + y = 0

2x - 3y = 0

• u = x + 4y
• u = 2x - 3y

J = ₂/^{11 4}/₁₁

V + 3y = 2U - 8y

1/11 ∫ ₁¹¹ ln(8u + 12v + 2u - v)/11

g'(x) = 1

l'(x) = 1/u + v

g(x) = u + v

[-x + v]

∫ ¹¹[(₀) - 17] ln(vt)]

• 12 ln(t_^t)
• = t 21 ln(x)
• - 17

t - 11

∫₀^v ln (v)

f'(x) = ln (v)

g'(x) = v

f(x) = ¹/_v

g(x) = ^v/₂

[^v2/₂ ln(v)]^v₀ - ∫₀^{v v}/₂ = ^v2/₂ ln (v) - ^v2/₂

¹/₁₀ [1² ln (v) - 1²/₂ ln (v) + ¹¹/₂ 11 + ¹¹/₂ ] = 11 ln (v) - 11 ln (v+1)

= ¹¹/₂ ln (v-1) + ¹¹/₄ + ¹¹/₂ - v

∫₁⁴ ln(2x+3y)

T = { 0,^A (1,3),^B (2,2) }

O^A y/3=x-1/x -y+3= -3x+3=3x-y=0 4√

O^B y/2=x-2/2 -2y+x=2x+x=2x-4y=0 -2x+2y=0

4√

• U=-3x-y
• V=-2x+2y

• x=u+y/3
• x=u+y/3

x=-u+4y

2u+4y = 3v+6y

4y=3v-2u

y=-3v-2u/4

x=4u-3v-2u/12

2u-3v/12

• U=3x-y
• V=-2x+2y

• y=3x-u
• y=v+2x/2

x=v+2u

y=3v+2u/4

J =

| -2 4 | 1/4

| 3 4 | 6/16 = 4/16 = 1/4

∫₁⁰ ln(4u+2v+2u+3v/4)

= (1/6)∫₀⁴ 6ln(6u+5v)-ln(x)

[ (6u+5v) ln(6u+5v) ]^4-v_-v - [

(24-6v+5v) ln(24-v) - ( 5v) ln(5v) -4+v]

(1/24)∫₀⁴ (2x-v) ln(2x-v) dv = ∫₂₀²⁴ f(x) ln(x) dx = -∫₂₄²⁰ (x) ln(x) dv

l(x)=ln (x)

l'(x) = 1/x

g⁻¹(x) = x

g⁻¹(x) = x²/2

[₃/² ln(x)]₂₀²⁴ - ∫₂₀²⁴ x/₂ dx

576/₂ ln(24) - 280 ln(20) - [_x²/⁴]₂₀²⁴

576/₂ ln(24) - 200 ln(20) - 576/₄ + 200

228 ln(24) - 200 ln(20) - 200 + 100

∫₀⁵v ln(5v)

f(u) = ln(5v)

g"_(u) = 5

g(x) = _5v²/²

∫₀^5v/₂ 5u/₂

(∫₀⁵v ln(5v))

∫₀⁴[40 ln(20) - ₅/ln(^v2/₂)]⁴

-40 ln(20) - - 20

1/24 [228 ln(24) - 200 ln(20) - 4x - 40 ln(20) + 20 - 16 + 8 - 16 ln(x) + 8 ln(x)]

19/₂ ln(x) - 85/₃ ln(20) - 4/₃ - 5/₃ ln(20) - ₇/³ ln(1) + 1/₃ ln(2)

-1,56

ϕ(t) = (t cos(t), t sin(t))

t ∈ [0, 10π]

Dϕ(t) = (-tsin(t)+cos(t), tcos(t)+sin(t))

∫ ||Dϕ(t)|| = 0^10π ∫ √ ...

∫₀^∞ t² + 1 dt ≈ 495

γ(t) = (t²cos(t), t²sin(t))

t ∈ ]0, 2π]

Dγ(t) = (2t cos(t) - t²sin(t) , 2t sin(t) + t²cos(t))

1 ∫ ln (x - y) dx dy

OA: y = -1/2 x + 4: x = -6/6 → -6y + 8 = -x + 8 → x - 6y = 0 → x + 6y = 8

OB: y = − x/2 + 4 → -4y + 8 = -2x + 8 → 2x - 4y = 0 = 8

{u = -x + 6y {x = -u + 6y {x = -u + 6y

{v = 2x - 4y {x = v + 4y/2 {-2u + 12y = v + 4y

x = 40 + 6v/8

y = v + 20/8

J = | 4/8 6/8 |

| 2/8 1 |

= 4/64 + 12/64 - 8/64 = 1/8

1/8 ∫₀^8-v ∫₀ ln (4v + 6v - v + 2v/8) du dv - 1/8 ∫₀^8-v ln (2u + 5v) -ln (8) du dv

1/2∫₀^8-v z ln (2u + 5v) du

f (x) = ln (2u + 5v) f' (x) = 2/2u + 5v

g' (x) = 2

g (x) = 8u + 5v

{[(2u + 5v) ln (2u + 5v)]₀^8-v -∫₀^8-v v u du

{ [ (8 - 2u + 5v) ln (16 + 3v) ] 0 -∫₀^8-v (5v) ln (5v)] - 8 + v-8 ln (8) + v ln (8)

{ (16 + 3v) ln (16 + 3v) dv = 1/36∫₀¹⁶ (x) ln (x) dx f (x) = ln (x) f ' (x) = 1/x

{ - ∫₁₆⁴⁰ ln (x)

{ 1/6 [1600 ln (40)-256 ln (16)] - 1/6 [x²/2]⁴⁰₁₆x²

[1600 ln (40) - 256 ln (8)] - ¹⁶⁰⁰/₁₂ + ²⁵⁶/₁₂

∫₀⁸ 5v ln (5v) dv = F(x) = ln (5v)   F'(x) = ¹/₅v   g'(x) = 5v g(x) = ^v2/₂

⁵/₂ [∫₀^v2/₂ (ln (5v)^v)⁸ dv = 160 ln (40) - ⁵/₂ [∫₀^v2/₂]⁸

160 ln (40) - 80

[¹⁶⁰⁰/₆ ln (40) - ²⁵⁶/₆ ln (16) - ¹⁶⁰⁰/₁₂ + ²⁵⁶/₁₂ = 260 ln (40) + 80 - 64 + 32 ln (8)]

+ 32 ln (8) ] ⁷/₁₆ = 9,04

φ(t) = (5 cos(t), sin(3t), 5 sin(t) + cos(3t))

t ∈ [0, 2π]

D_Y(t) = (-5 sin(t), 3 cos(3t), 5 cos(t) - 3 sin(3t))

|D_Y(t)| = ∫₀^2π √(25 sin²(t) + 9 cos²(3t) + 30 sin(t) cos(3t) + 25 cos²(t) + 9 - 36 cos(t) sin(3t)) dt

= ∫₀^2π √(25 - 25 cos²(t) + 15 sin²(t) + 45 sin²(2t) - 9 - 9 - 25 cos²(t) - 15 sin²(4t)) dt

= ∫₀^2π √(25 + 15 sin²(t) + 3 sin²(3t) - 9 - 15 sin²(t)) dt

= ∫₀^2π √(34 - 30 sin²(t)) - √(36 = ∫₀^2π √(1 - 30 sin²(t)) dt

= 34,31