integracion por partes ejemplo 3

00:12:47
https://www.youtube.com/watch?v=cyamDCLJfkE

Zusammenfassung

TLDRO vídeo aborda a resolución da integral de x multiplicada pola raíz cadrada de 3x + 2 usando a técnica de integración por partes. Este método involucra a elección dos termos a integrar e derivar. O vídeo guía o espectador na elección axeitada de u e dv, mostrando como derivar du e integrar para atopar v. Ao longo do proceso aplícanse outras técnicas, como a reescritura de raíces cadradas como expoñentes fraccionarios e a manipulación de termos algebraicos. Un punto crucial é a compensación polo factor de derivación ao integrar termos. Finalmente, o resultado final exprésase como unha expresión nela incluída unha constante de integración, reflectindo a solución xeral da integral dada.

Mitbringsel

  • 🔑 Uso da fórmula de integración por partes para resolver a integral
  • 📐 Selección de u = x e dv = raíz cadrada de 3x + 2 dx
  • 🧮 Derivación para obter du e integración para obter v
  • ➗ Manipulación e simplificación algébrica dos termos
  • 📏 Aplicación da fórmula de integración de potencias
  • 📝 Engadir a constante de integración ao resultado final
  • 📊 Convertendo raíces cadradas en forma expoñencial fraccionaria
  • 🔄 Compensación polo factor diferencial durante a integración
  • 🔧 Expresión final da integral inclúe termos expoñenciais e constantes
  • ⚖️ A importancia de precisar manipulacións algebraicas coidadosamente

Zeitleiste

  • 00:00:00 - 00:05:00

    Para resolver a integral da función x dividida pola raíz cadrada de 3x + 2, dx, utilizamos a fórmula de integración por partes: a integral de u dv é igual a u multiplicado por v menos a integral de v du. Escollemos u como x e dv como a raíz cadrada de 3x + 2, dx. Calculamos a derivada para obter du e integramos para obter v. A raíz cadrada convértese nunha potencia para integrar usando a fórmula: a integral de u elevado a n, du é igual a u elevado a n+1 dividido por n+1, máis c, compensando o multiplicador engadido ao diferencial para completar a forma diferenciable.

  • 00:05:00 - 00:12:47

    Unha vez obtidos u, du, v e confirmado o diferencial, aplicamos a fórmula de integración por partes: a integral de x raíz cadrada de 3x + 2, dx, é igual a u multiplicado por v menos a integral de v du. Simplificamos os termos, integramos a potencia restante de 3x + 2 constante n = 3/2 tras completar o diferencial necesario, e rematamos engadindo a constante de integración c. O resultado final en forma simplificada implica facer cálculos adicionais para reducir as fraccións a súa forma máis sinxela e a conversión a raíces cadradas para expresar o resultado axeitadamente.

Mind Map

Mind Map

Video-Fragen und Antworten

  • Que método se usa para resolver a integral?

    Úsase o método de integración por partes para resolver a integral.

  • Cal é a fórmula básica de integración por partes?

    A fórmula básica é ∫u dv = u*v - ∫v du.

  • Como se escolle u e dv?

    No vídeo escóllese u = x e dv = raíz cadrada de 3x + 2 dx.

  • Que deriva do termo u = x?

    A derivada de u = x é dx e isto dá lugar a du = dx.

  • Como se resolve a integral da raíz cadrada?

    A raíz cadrada exprésase como unha potencia (3x + 2)^(1/2) e intégrase usando a fórmula para a integración de potencias.

  • Que se fai ao integrar por partes con v?

    Vénse que v é derivado integrando dv e manipúlase algebraicamente para poder aplicar a fórmula de integración por partes.

  • Por que se engade unha constante de integración?

    A constante de integración c engádese nos resultados finais para acomodar calquera constante orixinada polo proceso de integración.

  • Cal é a forma final da integral resolta?

    A integral resolta exprésase como 2/9 x (3x + 2)^(3/2) - 4/135 (3x + 2)^(5/2) + c.

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Untertitel
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Automatisches Blättern:
  • 00:00:01
    We want to get the integral that we have here.
  • 00:00:05
    The integral of x by the square root of 3x + 2, dx.
  • 00:00:11
    We want to use the integration formula by parts, which we have here.
  • 00:00:16
    The integral of u dv is equal to u multiplied by v minus the integral of v du.
  • 00:00:25
    The formula of integration by parts requires choosing u and dv of the integral that we have.
  • 00:00:34
    Let's choose u and dv.
  • 00:00:49
    Let's take u as x ...
  • 00:00:52
    ... and dv as the square root of 3x + 2, dx.
  • 00:01:07
    dv should always carry dx.
  • 00:01:17
    From here we must obtain du and from here we must obtain v.
  • 00:01:21
    Here we must derive and here we must integrate.
  • 00:01:26
    Let's get du from this.
  • 00:01:32
    du is the derivative of this, multiplied by dx.
  • 00:01:36
    The derivative of this x is 1.
  • 00:01:39
    1 multiplied by dx is equal to dx.
  • 00:01:43
    And that is equal to du.
  • 00:01:46
    Here we must integrate both sides of the equation.
  • 00:01:55
    Here the integral with the differential are canceled.
  • 00:02:00
    And only the variable remains.
  • 00:02:07
    We have the integral of a square root.
  • 00:02:10
    This square root can be integrated as a power.
  • 00:02:15
    Because in fact, it can be written as a power. It can be written as 3x + 2 at 1/2.
  • 00:02:26
    And there is a formula to integrate a power, which is this.
  • 00:02:56
    The integral of u to n, du is equal to a quotient. Above we put u to n + 1, divided by n + 1, plus c.
  • 00:03:07
    This is of this type. This is the function u. This is exponent n.
  • 00:03:12
    This must be the du, (according to this formula).
  • 00:03:16
    u
  • 00:03:17
    u would be 3x + 2, n would be 1/2, (a constant), and this should be du.
  • 00:03:25
    If this is u,
  • 00:03:27
    3x + 2.
  • 00:03:30
    The derivative is 3. That is, the differential would be 3dx.
  • 00:03:35
    So, for this to be du, we must multiply by 3. We will do it.
  • 00:03:41
    Multiply by 3. To have 3dx.
  • 00:03:44
    That is, we want this to be du.
  • 00:03:46
    u to n multiplied by du.
  • 00:03:48
    but this 3 changes the situation. We must compensate, dividing by 3.
  • 00:03:53
    All ready, we can use this formula.
  • 00:03:57
    The integral of u to n, multiplied by du.
  • 00:04:00
    u to n
  • 00:04:01
    and this is the du of this u.
  • 00:04:06
    We are going to put the result.
  • 00:04:09
    One third multiplied by
  • 00:04:12
    what the formula says, which is
  • 00:04:16
    is a quotient: we put the u above, which is
  • 00:04:19
    3x + 2.
  • 00:04:24
    to
  • 00:04:26
    n + 1, that is we add 1 to the exponent.
  • 00:04:29
    if we add 1 to 1/2, we are adding 2/2.
  • 00:04:32
    then, we would get 3/2.
  • 00:04:35
    3/2
  • 00:04:38
    divided by 3/2.
  • 00:04:41
    In other words, the same. The same here as here.
  • 00:04:44
    We should add c. But c is not written when v is calculated in the integration by parts.
  • 00:04:52
    This v would give us (I'll put it here).
  • 00:04:59
    Here we are going to multiply the extremes and the means.
  • 00:05:07
    1 multiplied by 2 gives 2. Above.
  • 00:05:10
    divided by 3 multiplied by 3, which gives 9. Down. 2/9.
  • 00:05:17
    And this, we can put it here. As well as power, or as a root.
  • 00:05:24
    Let's put it as it is.
  • 00:05:36
    We already calculate v, then we can apply the integration formula by parts. Let's put it here.
  • 00:05:46
    It means that this integral, the problem, the integral of x root of 3x + 2,
  • 00:05:57
    dx, is equal to:
  • 00:06:00
    u multiplied by v.
  • 00:06:03
    u multiplied by v.
  • 00:06:05
    This, which is u. That is, x.
  • 00:06:09
    Multiplied by v, that is, by this.
  • 00:06:12
    Multiplied by 2/9.
  • 00:06:15
    That is, x multiplied by 2/9.
  • 00:06:20
    of 3x + 2 to 3/2.
  • 00:06:29
    u multiplied by v ... minus
  • 00:06:32
    minus
  • 00:06:36
    The integral of v du.
  • 00:06:39
    The integral of v. It is this.
  • 00:06:44
    2/9 of ...
  • 00:06:53
    This is v.
  • 00:06:55
    Multiplied by du.
  • 00:06:57
    du is dx.
  • 00:07:06
    Now let's simplify this.
  • 00:07:10
    It would be 2/9 ...
  • 00:07:16
    of x ...
  • 00:07:19
    This power ...
  • 00:07:26
    minus ...
  • 00:07:29
    This goes out of the integral.
  • 00:07:54
    Now we must obtain this integral.
  • 00:08:00
    It is the integral of a power.
  • 00:08:04
    That has as base 3x + 2 and the constant exponent 3/2.
  • 00:08:12
    We can obtain it with this formula: the integral of u to n, du.
  • 00:08:17
    It is obvious that u would be 3x + 2, n would be 3/2.
  • 00:08:22
    We must complete the differential.
  • 00:08:25
    The derivative of 3x + 2 is 3.
  • 00:08:28
    Then, the differential would be 3dx.
  • 00:08:31
    I will place here a 3 to complete it.
  • 00:08:36
    But I must divide by 3 here.
  • 00:08:39
    I will place here a 3, which will accompany the 9.
  • 00:08:47
    That is, if there is multiplied by 3, here it should be divided by 3, so that it is compensated.
  • 00:08:52
    So I have u to n, du.
  • 00:08:55
    That is, this 3 is the derivative of 3x + 2. I have 3dx. u to n, du. I can apply the formula, now.
  • 00:09:05
    Let's do it here. Here I will put the result.
  • 00:09:10
    In other words, the step that follows here below, I will put it here.
  • 00:09:16
    This is going to be put here.
  • 00:09:20
    2/9 of x multiplied by
  • 00:09:27
    3x + 2
  • 00:09:32
    to 3/2
  • 00:09:36
    minus
  • 00:09:40
    2 divided by 3 multiplied by 9, which is 2 divided by 27.
  • 00:09:50
    Now, this integral: the integral of u to n, du. As it is already complete, here is the result. Let's put it there.
  • 00:10:00
    u which is the base, which is 3x + 2.
  • 00:10:09
    To the exponent I add 1, that is, to n, which is 3/2, I add 1 to it.
  • 00:10:14
    3/2 + 2/2 would be. It turns out: 5/2. I put it here.
  • 00:10:24
    divided by the same. Here.
  • 00:10:28
    The result here is the same: 5/2.
  • 00:10:32
    + c
  • 00:10:34
    Now I do add an integration constant.
  • 00:10:48
    Now let's simplify these fractions.
  • 00:10:59
    Let's continue here.
  • 00:11:02
    two ...
  • 00:11:05
    ninths
  • 00:11:08
    of x.
  • 00:11:14
    This can be written as root. It is the square root of 3x + 2, to the cubic power.
  • 00:11:30
    minus
  • 00:11:34
    Here we are going to make a multiplication ... It is a fraction divided by another ... Let's multiply the extremes and the means.
  • 00:11:43
    2 multiplied by 2 is 4 (we wrote it above). Divided by 27 multiplied by 5 (we write it below). 27 multiplied by 5 is ...
  • 00:11:57
    135
  • 00:12:05
    And if we turn this into a root, we have: square root of 3x + 2,
  • 00:12:18
    to the fifth power.
  • 00:12:23
    + c
  • 00:12:34
    We are going to leave the result in this way.
Tags
  • integración
  • cálculo
  • matemáticas
  • integración por partes
  • áxebra
  • raíces cadradas
  • expoñentes
  • derivadas
  • fórmulas
  • constante de integración