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Resolución de integrales/ (x-4)/ 4-(sqrt(16-((x-4)^2)))

Integral de 4-(sqrt(16-((x-4)^2))) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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 |  \4 - \/  16 - (x - 4)  / dx
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0
∫01(4−16−(x−4)2) dx\int\limits_{0}^{1} \left(4 - \sqrt{16 - \left(x - 4\right)^{2}}\right)\, dx 
Integral(4 - sqrt(16 - (x - 4)^2), (x, 0, 1))
Solución detallada

  1. Integramos término a término:

    1. La integral de las constantes tienen esta constante multiplicada por la variable de integración:

      ∫4 dx=4x\int 4\, dx = 4 x

    1. La integral del producto de una función por una constante es la constante por la integral de esta función:

      ∫(−16−(x−4)2) dx=−∫16−(x−4)2 dx\int \left(- \sqrt{16 - \left(x - 4\right)^{2}}\right)\, dx = - \int \sqrt{16 - \left(x - 4\right)^{2}}\, dx

      1. No puedo encontrar los pasos en la búsqueda de esta integral.

        Pero la integral

        {i(x−4)32(x−4)2−16−8i(x−4)(x−4)2−16−8iacosh⁡(x4−1)for ∣(x−4)2∣16>18asin⁡(x4−1)−(x−4)3216−(x−4)2+8(x−4)16−(x−4)2otherwese\begin{cases} \frac{i \left(x - 4\right)^{3}}{2 \sqrt{\left(x - 4\right)^{2} - 16}} - \frac{8 i \left(x - 4\right)}{\sqrt{\left(x - 4\right)^{2} - 16}} - 8 i \operatorname{acosh}{\left(\frac{x}{4} - 1 \right)} & \text{for}\: \frac{\left|{\left(x - 4\right)^{2}}\right|}{16} > 1 \\8 \operatorname{asin}{\left(\frac{x}{4} - 1 \right)} - \frac{\left(x - 4\right)^{3}}{2 \sqrt{16 - \left(x - 4\right)^{2}}} + \frac{8 \left(x - 4\right)}{\sqrt{16 - \left(x - 4\right)^{2}}} & \text{otherwese} \end{cases}

      Por lo tanto, el resultado es: −{i(x−4)32(x−4)2−16−8i(x−4)(x−4)2−16−8iacosh⁡(x4−1)for ∣(x−4)2∣16>18asin⁡(x4−1)−(x−4)3216−(x−4)2+8(x−4)16−(x−4)2otherwese- \begin{cases} \frac{i \left(x - 4\right)^{3}}{2 \sqrt{\left(x - 4\right)^{2} - 16}} - \frac{8 i \left(x - 4\right)}{\sqrt{\left(x - 4\right)^{2} - 16}} - 8 i \operatorname{acosh}{\left(\frac{x}{4} - 1 \right)} & \text{for}\: \frac{\left|{\left(x - 4\right)^{2}}\right|}{16} > 1 \\8 \operatorname{asin}{\left(\frac{x}{4} - 1 \right)} - \frac{\left(x - 4\right)^{3}}{2 \sqrt{16 - \left(x - 4\right)^{2}}} + \frac{8 \left(x - 4\right)}{\sqrt{16 - \left(x - 4\right)^{2}}} & \text{otherwese} \end{cases}

    El resultado es: 4x−{i(x−4)32(x−4)2−16−8i(x−4)(x−4)2−16−8iacosh⁡(x4−1)for ∣(x−4)2∣16>18asin⁡(x4−1)−(x−4)3216−(x−4)2+8(x−4)16−(x−4)2otherwese4 x - \begin{cases} \frac{i \left(x - 4\right)^{3}}{2 \sqrt{\left(x - 4\right)^{2} - 16}} - \frac{8 i \left(x - 4\right)}{\sqrt{\left(x - 4\right)^{2} - 16}} - 8 i \operatorname{acosh}{\left(\frac{x}{4} - 1 \right)} & \text{for}\: \frac{\left|{\left(x - 4\right)^{2}}\right|}{16} > 1 \\8 \operatorname{asin}{\left(\frac{x}{4} - 1 \right)} - \frac{\left(x - 4\right)^{3}}{2 \sqrt{16 - \left(x - 4\right)^{2}}} + \frac{8 \left(x - 4\right)}{\sqrt{16 - \left(x - 4\right)^{2}}} & \text{otherwese} \end{cases}

  2. Ahora simplificar:

    {8xx(x−8)+i(−x3+12x2−32x+16x(x−8)acosh⁡(x4−1))2x(x−8)for ∣(x−4)2∣16>1x3−12x2+8xx(8−x)+32x−16x(8−x)asin⁡(x4−1)2x(8−x)otherwese\begin{cases} \frac{8 x \sqrt{x \left(x - 8\right)} + i \left(- x^{3} + 12 x^{2} - 32 x + 16 \sqrt{x \left(x - 8\right)} \operatorname{acosh}{\left(\frac{x}{4} - 1 \right)}\right)}{2 \sqrt{x \left(x - 8\right)}} & \text{for}\: \frac{\left|{\left(x - 4\right)^{2}}\right|}{16} > 1 \\\frac{x^{3} - 12 x^{2} + 8 x \sqrt{x \left(8 - x\right)} + 32 x - 16 \sqrt{x \left(8 - x\right)} \operatorname{asin}{\left(\frac{x}{4} - 1 \right)}}{2 \sqrt{x \left(8 - x\right)}} & \text{otherwese} \end{cases}

  3. Añadimos la constante de integración:

    {8xx(x−8)+i(−x3+12x2−32x+16x(x−8)acosh⁡(x4−1))2x(x−8)for ∣(x−4)2∣16>1x3−12x2+8xx(8−x)+32x−16x(8−x)asin⁡(x4−1)2x(8−x)otherwese+constant\begin{cases} \frac{8 x \sqrt{x \left(x - 8\right)} + i \left(- x^{3} + 12 x^{2} - 32 x + 16 \sqrt{x \left(x - 8\right)} \operatorname{acosh}{\left(\frac{x}{4} - 1 \right)}\right)}{2 \sqrt{x \left(x - 8\right)}} & \text{for}\: \frac{\left|{\left(x - 4\right)^{2}}\right|}{16} > 1 \\\frac{x^{3} - 12 x^{2} + 8 x \sqrt{x \left(8 - x\right)} + 32 x - 16 \sqrt{x \left(8 - x\right)} \operatorname{asin}{\left(\frac{x}{4} - 1 \right)}}{2 \sqrt{x \left(8 - x\right)}} & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta:

{8xx(x−8)+i(−x3+12x2−32x+16x(x−8)acosh⁡(x4−1))2x(x−8)for ∣(x−4)2∣16>1x3−12x2+8xx(8−x)+32x−16x(8−x)asin⁡(x4−1)2x(8−x)otherwese+constant\begin{cases} \frac{8 x \sqrt{x \left(x - 8\right)} + i \left(- x^{3} + 12 x^{2} - 32 x + 16 \sqrt{x \left(x - 8\right)} \operatorname{acosh}{\left(\frac{x}{4} - 1 \right)}\right)}{2 \sqrt{x \left(x - 8\right)}} & \text{for}\: \frac{\left|{\left(x - 4\right)^{2}}\right|}{16} > 1 \\\frac{x^{3} - 12 x^{2} + 8 x \sqrt{x \left(8 - x\right)} + 32 x - 16 \sqrt{x \left(8 - x\right)} \operatorname{asin}{\left(\frac{x}{4} - 1 \right)}}{2 \sqrt{x \left(8 - x\right)}} & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                     //                                     3                                   |        2|    \      
                                     ||           /     x\        I*(-4 + x)              8*I*(-4 + x)          |(-4 + x) |    |      
  /                                  ||- 8*I*acosh|-1 + -| + ---------------------- - --------------------  for ----------- > 1|      
 |                                   ||           \     4/        _________________      _________________           16        |      
 | /       _______________\          ||                          /               2      /               2                      |      
 | |      /             2 |          ||                      2*\/  -16 + (-4 + x)     \/  -16 + (-4 + x)                       |      
 | \4 - \/  16 - (x - 4)  / dx = C - |<                                                                                        | + 4*x
 |                                   ||                                                        3                               |      
/                                    ||         /     x\        8*(-4 + x)             (-4 + x)                                |      
                                     ||   8*asin|-1 + -| + ------------------- - ---------------------           otherwise     |      
                                     ||         \     4/      ________________        ________________                         |      
                                     ||                      /              2        /              2                          |      
                                     \\                    \/  16 - (-4 + x)     2*\/  16 - (-4 + x)                           /
∫(4−16−(x−4)2) dx=C+4x−{i(x−4)32(x−4)2−16−8i(x−4)(x−4)2−16−8iacosh⁡(x4−1)for ∣(x−4)2∣16>18asin⁡(x4−1)−(x−4)3216−(x−4)2+8(x−4)16−(x−4)2otherwise\int \left(4 - \sqrt{16 - \left(x - 4\right)^{2}}\right)\, dx = C + 4 x - \begin{cases} \frac{i \left(x - 4\right)^{3}}{2 \sqrt{\left(x - 4\right)^{2} - 16}} - \frac{8 i \left(x - 4\right)}{\sqrt{\left(x - 4\right)^{2} - 16}} - 8 i \operatorname{acosh}{\left(\frac{x}{4} - 1 \right)} & \text{for}\: \frac{\left|{\left(x - 4\right)^{2}}\right|}{16} > 1 \\8 \operatorname{asin}{\left(\frac{x}{4} - 1 \right)} - \frac{\left(x - 4\right)^{3}}{2 \sqrt{16 - \left(x - 4\right)^{2}}} + \frac{8 \left(x - 4\right)}{\sqrt{16 - \left(x - 4\right)^{2}}} & \text{otherwise} \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90020
Trazar el gráfico f(x)
Respuesta [src] 
      1                                                                                                                                             
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     |  /                                                               2                                          3                        2       
     |  |         2*I                    8*I                3*I*(-4 + x)         8*I*(-4 + x)*(4 - x)    I*(-4 + x) *(4 - x)        (-4 + x)        
     |  |--------------------- + -------------------- - ---------------------- + -------------------- - ----------------------  for --------- > 1   
     |  |     ________________      _________________        _________________                    3/2                      3/2          16          
     |  |    /              2      /               2        /               2    /              2\        /              2\                         
     |  |   /       /     x\     \/  -16 + (-4 + x)     2*\/  -16 + (-4 + x)     \-16 + (-4 + x) /      2*\-16 + (-4 + x) /                         
     |  |  /   -1 + |-1 + -|                                                                                                                        
     |  |\/         \     4/                                                                                                                        
4 +  |  <                                                                                                                                         dx
     |  |                                                              4                       2                      2                             
     |  |            8                     2                   (-4 + x)              8*(-4 + x)             3*(-4 + x)                              
     |  | - ------------------- - -------------------- + --------------------- - ------------------- + ---------------------        otherwise       
     |  |      ________________        _______________                     3/2                   3/2        ________________                        
     |  |     /              2        /             2      /             2\      /             2\          /              2                         
     |  |   \/  16 - (-4 + x)        /      /     x\     2*\16 - (-4 + x) /      \16 - (-4 + x) /      2*\/  16 - (-4 + x)                          
     |  |                           /   1 - |-1 + -|                                                                                                
     |  \                         \/        \     4/                                                                                                
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    0
∫01{−i(4−x)(x−4)32((x−4)2−16)32+8i(4−x)(x−4)((x−4)2−16)32−3i(x−4)22(x−4)2−16+8i(x−4)2−16+2i(x4−1)2−1for (x−4)216>13(x−4)2216−(x−4)2−816−(x−4)2+(x−4)42(16−(x−4)2)32−8(x−4)2(16−(x−4)2)32−21−(x4−1)2otherwise dx+4\int\limits_{0}^{1} \begin{cases} - \frac{i \left(4 - x\right) \left(x - 4\right)^{3}}{2 \left(\left(x - 4\right)^{2} - 16\right)^{\frac{3}{2}}} + \frac{8 i \left(4 - x\right) \left(x - 4\right)}{\left(\left(x - 4\right)^{2} - 16\right)^{\frac{3}{2}}} - \frac{3 i \left(x - 4\right)^{2}}{2 \sqrt{\left(x - 4\right)^{2} - 16}} + \frac{8 i}{\sqrt{\left(x - 4\right)^{2} - 16}} + \frac{2 i}{\sqrt{\left(\frac{x}{4} - 1\right)^{2} - 1}} & \text{for}\: \frac{\left(x - 4\right)^{2}}{16} > 1 \\\frac{3 \left(x - 4\right)^{2}}{2 \sqrt{16 - \left(x - 4\right)^{2}}} - \frac{8}{\sqrt{16 - \left(x - 4\right)^{2}}} + \frac{\left(x - 4\right)^{4}}{2 \left(16 - \left(x - 4\right)^{2}\right)^{\frac{3}{2}}} - \frac{8 \left(x - 4\right)^{2}}{\left(16 - \left(x - 4\right)^{2}\right)^{\frac{3}{2}}} - \frac{2}{\sqrt{1 - \left(\frac{x}{4} - 1\right)^{2}}} & \text{otherwise} \end{cases}\, dx + 4 
=
= 
      1                                                                                                                                             
      /                                                                                                                                             
     |                                                                                                                                              
     |  /                                                               2                                          3                        2       
     |  |         2*I                    8*I                3*I*(-4 + x)         8*I*(-4 + x)*(4 - x)    I*(-4 + x) *(4 - x)        (-4 + x)        
     |  |--------------------- + -------------------- - ---------------------- + -------------------- - ----------------------  for --------- > 1   
     |  |     ________________      _________________        _________________                    3/2                      3/2          16          
     |  |    /              2      /               2        /               2    /              2\        /              2\                         
     |  |   /       /     x\     \/  -16 + (-4 + x)     2*\/  -16 + (-4 + x)     \-16 + (-4 + x) /      2*\-16 + (-4 + x) /                         
     |  |  /   -1 + |-1 + -|                                                                                                                        
     |  |\/         \     4/                                                                                                                        
4 +  |  <                                                                                                                                         dx
     |  |                                                              4                       2                      2                             
     |  |            8                     2                   (-4 + x)              8*(-4 + x)             3*(-4 + x)                              
     |  | - ------------------- - -------------------- + --------------------- - ------------------- + ---------------------        otherwise       
     |  |      ________________        _______________                     3/2                   3/2        ________________                        
     |  |     /              2        /             2      /             2\      /             2\          /              2                         
     |  |   \/  16 - (-4 + x)        /      /     x\     2*\16 - (-4 + x) /      \16 - (-4 + x) /      2*\/  16 - (-4 + x)                          
     |  |                           /   1 - |-1 + -|                                                                                                
     |  \                         \/        \     4/                                                                                                
     |                                                                                                                                              
    /                                                                                                                                               
    0
∫01{−i(4−x)(x−4)32((x−4)2−16)32+8i(4−x)(x−4)((x−4)2−16)32−3i(x−4)22(x−4)2−16+8i(x−4)2−16+2i(x4−1)2−1for (x−4)216>13(x−4)2216−(x−4)2−816−(x−4)2+(x−4)42(16−(x−4)2)32−8(x−4)2(16−(x−4)2)32−21−(x4−1)2otherwise dx+4\int\limits_{0}^{1} \begin{cases} - \frac{i \left(4 - x\right) \left(x - 4\right)^{3}}{2 \left(\left(x - 4\right)^{2} - 16\right)^{\frac{3}{2}}} + \frac{8 i \left(4 - x\right) \left(x - 4\right)}{\left(\left(x - 4\right)^{2} - 16\right)^{\frac{3}{2}}} - \frac{3 i \left(x - 4\right)^{2}}{2 \sqrt{\left(x - 4\right)^{2} - 16}} + \frac{8 i}{\sqrt{\left(x - 4\right)^{2} - 16}} + \frac{2 i}{\sqrt{\left(\frac{x}{4} - 1\right)^{2} - 1}} & \text{for}\: \frac{\left(x - 4\right)^{2}}{16} > 1 \\\frac{3 \left(x - 4\right)^{2}}{2 \sqrt{16 - \left(x - 4\right)^{2}}} - \frac{8}{\sqrt{16 - \left(x - 4\right)^{2}}} + \frac{\left(x - 4\right)^{4}}{2 \left(16 - \left(x - 4\right)^{2}\right)^{\frac{3}{2}}} - \frac{8 \left(x - 4\right)^{2}}{\left(16 - \left(x - 4\right)^{2}\right)^{\frac{3}{2}}} - \frac{2}{\sqrt{1 - \left(\frac{x}{4} - 1\right)^{2}}} & \text{otherwise} \end{cases}\, dx + 4 
4 + Integral(Piecewise((2*i/sqrt(-1 + (-1 + x/4)^2) + 8*i/sqrt(-16 + (-4 + x)^2) - 3*i*(-4 + x)^2/(2*sqrt(-16 + (-4 + x)^2)) + 8*i*(-4 + x)*(4 - x)/(-16 + (-4 + x)^2)^(3/2) - i*(-4 + x)^3*(4 - x)/(2*(-16 + (-4 + x)^2)^(3/2)), (-4 + x)^2/16 > 1), (-8/sqrt(16 - (-4 + x)^2) - 2/sqrt(1 - (-1 + x/4)^2) + (-4 + x)^4/(2*(16 - (-4 + x)^2)^(3/2)) - 8*(-4 + x)^2/(16 - (-4 + x)^2)^(3/2) + 3*(-4 + x)^2/(2*sqrt(16 - (-4 + x)^2)), True)), (x, 0, 1))
Respuesta numérica [src] 
2.18675298408956
2.18675298408956

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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