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Resolución de integrales/ arctan/ sqrt(x)/ arctan(sqrt(sqrt(x)-1))

Integral de arctan(sqrt(sqrt(x)-1)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 16                        
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 |  atan\\/  \/ x  - 1 / dx
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1
∫116atan⁡(x−1) dx\int\limits_{1}^{16} \operatorname{atan}{\left(\sqrt{\sqrt{x} - 1} \right)}\, dx 
Integral(atan(sqrt(sqrt(x) - 1)), (x, 1, 16))
Solución detallada

  1. Usamos la integración por partes:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    que u(x)=atan⁡(x−1)u{\left(x \right)} = \operatorname{atan}{\left(\sqrt{\sqrt{x} - 1} \right)} y que dv⁡(x)=1\operatorname{dv}{\left(x \right)} = 1.

    Entonces du⁡(x)=14xx−1\operatorname{du}{\left(x \right)} = \frac{1}{4 x \sqrt{\sqrt{x} - 1}}.

    Para buscar v(x)v{\left(x \right)}:

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

      ∫1 dx=x\int 1\, dx = x

    Ahora resolvemos podintegral.

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

    ∫14x−1 dx=∫1x−1 dx4\int \frac{1}{4 \sqrt{\sqrt{x} - 1}}\, dx = \frac{\int \frac{1}{\sqrt{\sqrt{x} - 1}}\, dx}{4}

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

      Pero la integral

      {−4x52x−1−3x52+3x2+8ix52−3x52+3x2−4x3x−1−3x52+3x2+8x2x−1−3x52+3x2−8ix2−3x52+3x2for ∣x∣>1−4ix521−x−3x52+3x2+8ix52−3x52+3x2−4ix31−x−3x52+3x2+8ix21−x−3x52+3x2−8ix2−3x52+3x2otherwese\begin{cases} - \frac{4 x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 i x^{\frac{5}{2}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{4 x^{3} \sqrt{\sqrt{x} - 1}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 x^{2} \sqrt{\sqrt{x} - 1}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{8 i x^{2}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} & \text{for}\: \left|{\sqrt{x}}\right| > 1 \\- \frac{4 i x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 i x^{\frac{5}{2}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{4 i x^{3} \sqrt{1 - \sqrt{x}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 i x^{2} \sqrt{1 - \sqrt{x}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{8 i x^{2}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} & \text{otherwese} \end{cases}

    Por lo tanto, el resultado es: {−4x52x−1−3x52+3x2+8ix52−3x52+3x2−4x3x−1−3x52+3x2+8x2x−1−3x52+3x2−8ix2−3x52+3x2for ∣x∣>1−4ix521−x−3x52+3x2+8ix52−3x52+3x2−4ix31−x−3x52+3x2+8ix21−x−3x52+3x2−8ix2−3x52+3x2otherwese4\frac{\begin{cases} - \frac{4 x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 i x^{\frac{5}{2}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{4 x^{3} \sqrt{\sqrt{x} - 1}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 x^{2} \sqrt{\sqrt{x} - 1}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{8 i x^{2}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} & \text{for}\: \left|{\sqrt{x}}\right| > 1 \\- \frac{4 i x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 i x^{\frac{5}{2}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{4 i x^{3} \sqrt{1 - \sqrt{x}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 i x^{2} \sqrt{1 - \sqrt{x}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{8 i x^{2}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} & \text{otherwese} \end{cases}}{4}

  3. Ahora simplificar:

    {−x52x−1+2ix52−x3x−1+2x2x−1−2ix2+3x(x52−x2)atan⁡(x−1)3(x52−x2)for ∣x∣>1x(x52−x2)atan⁡(x−1)+i(−x521−x+2x52−x31−x+2x21−x−2x2)3x52−x2otherwese\begin{cases} \frac{- x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1} + 2 i x^{\frac{5}{2}} - x^{3} \sqrt{\sqrt{x} - 1} + 2 x^{2} \sqrt{\sqrt{x} - 1} - 2 i x^{2} + 3 x \left(x^{\frac{5}{2}} - x^{2}\right) \operatorname{atan}{\left(\sqrt{\sqrt{x} - 1} \right)}}{3 \left(x^{\frac{5}{2}} - x^{2}\right)} & \text{for}\: \left|{\sqrt{x}}\right| > 1 \\\frac{x \left(x^{\frac{5}{2}} - x^{2}\right) \operatorname{atan}{\left(\sqrt{\sqrt{x} - 1} \right)} + \frac{i \left(- x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 2 x^{\frac{5}{2}} - x^{3} \sqrt{1 - \sqrt{x}} + 2 x^{2} \sqrt{1 - \sqrt{x}} - 2 x^{2}\right)}{3}}{x^{\frac{5}{2}} - x^{2}} & \text{otherwese} \end{cases}

  4. Añadimos la constante de integración:

    {−x52x−1+2ix52−x3x−1+2x2x−1−2ix2+3x(x52−x2)atan⁡(x−1)3(x52−x2)for ∣x∣>1x(x52−x2)atan⁡(x−1)+i(−x521−x+2x52−x31−x+2x21−x−2x2)3x52−x2otherwese+constant\begin{cases} \frac{- x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1} + 2 i x^{\frac{5}{2}} - x^{3} \sqrt{\sqrt{x} - 1} + 2 x^{2} \sqrt{\sqrt{x} - 1} - 2 i x^{2} + 3 x \left(x^{\frac{5}{2}} - x^{2}\right) \operatorname{atan}{\left(\sqrt{\sqrt{x} - 1} \right)}}{3 \left(x^{\frac{5}{2}} - x^{2}\right)} & \text{for}\: \left|{\sqrt{x}}\right| > 1 \\\frac{x \left(x^{\frac{5}{2}} - x^{2}\right) \operatorname{atan}{\left(\sqrt{\sqrt{x} - 1} \right)} + \frac{i \left(- x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 2 x^{\frac{5}{2}} - x^{3} \sqrt{1 - \sqrt{x}} + 2 x^{2} \sqrt{1 - \sqrt{x}} - 2 x^{2}\right)}{3}}{x^{\frac{5}{2}} - x^{2}} & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta:

{−x52x−1+2ix52−x3x−1+2x2x−1−2ix2+3x(x52−x2)atan⁡(x−1)3(x52−x2)for ∣x∣>1x(x52−x2)atan⁡(x−1)+i(−x521−x+2x52−x31−x+2x21−x−2x2)3x52−x2otherwese+constant\begin{cases} \frac{- x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1} + 2 i x^{\frac{5}{2}} - x^{3} \sqrt{\sqrt{x} - 1} + 2 x^{2} \sqrt{\sqrt{x} - 1} - 2 i x^{2} + 3 x \left(x^{\frac{5}{2}} - x^{2}\right) \operatorname{atan}{\left(\sqrt{\sqrt{x} - 1} \right)}}{3 \left(x^{\frac{5}{2}} - x^{2}\right)} & \text{for}\: \left|{\sqrt{x}}\right| > 1 \\\frac{x \left(x^{\frac{5}{2}} - x^{2}\right) \operatorname{atan}{\left(\sqrt{\sqrt{x} - 1} \right)} + \frac{i \left(- x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 2 x^{\frac{5}{2}} - x^{3} \sqrt{1 - \sqrt{x}} + 2 x^{2} \sqrt{1 - \sqrt{x}} - 2 x^{2}\right)}{3}}{x^{\frac{5}{2}} - x^{2}} & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                 /                             ____________             ____________                             ____________                                            
                                 |             2          3   /        ___       5/2   /        ___             5/2         2   /        ___                                             
                                 |        8*I*x        4*x *\/  -1 + \/ x     4*x   *\/  -1 + \/ x         8*I*x         8*x *\/  -1 + \/ x          |  ___|                             
                                 | - --------------- - -------------------- - ---------------------- + --------------- + --------------------    for |\/ x | > 1                         
                                 |        5/2      2          5/2      2              5/2      2            5/2      2          5/2      2                                               
                                 |   - 3*x    + 3*x      - 3*x    + 3*x          - 3*x    + 3*x        - 3*x    + 3*x      - 3*x    + 3*x                                                
                                 <                                                                                                                                                       
                                 |                                                ___________               ___________             ___________                                          
                                 |            2                5/2           3   /       ___         5/2   /       ___         2   /       ___                                           
  /                              |       8*I*x            8*I*x         4*I*x *\/  1 - \/ x     4*I*x   *\/  1 - \/ x     8*I*x *\/  1 - \/ x                                            
 |                               |- --------------- + --------------- - --------------------- - ----------------------- + ---------------------     otherwise                            
 |     /   ___________\          |       5/2      2        5/2      2           5/2      2               5/2      2               5/2      2                             /   ___________\
 |     |  /   ___     |          \  - 3*x    + 3*x    - 3*x    + 3*x       - 3*x    + 3*x           - 3*x    + 3*x           - 3*x    + 3*x                              |  /   ___     |
 | atan\\/  \/ x  - 1 / dx = C - ------------------------------------------------------------------------------------------------------------------------------- + x*atan\\/  \/ x  - 1 /
 |                                                                                              4                                                                                        
/
∫atan⁡(x−1) dx=C+xatan⁡(x−1)−{−4x52x−1−3x52+3x2+8ix52−3x52+3x2−4x3x−1−3x52+3x2+8x2x−1−3x52+3x2−8ix2−3x52+3x2for ∣x∣>1−4ix521−x−3x52+3x2+8ix52−3x52+3x2−4ix31−x−3x52+3x2+8ix21−x−3x52+3x2−8ix2−3x52+3x2otherwise4\int \operatorname{atan}{\left(\sqrt{\sqrt{x} - 1} \right)}\, dx = C + x \operatorname{atan}{\left(\sqrt{\sqrt{x} - 1} \right)} - \frac{\begin{cases} - \frac{4 x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 i x^{\frac{5}{2}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{4 x^{3} \sqrt{\sqrt{x} - 1}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 x^{2} \sqrt{\sqrt{x} - 1}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{8 i x^{2}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} & \text{for}\: \left|{\sqrt{x}}\right| > 1 \\- \frac{4 i x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 i x^{\frac{5}{2}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{4 i x^{3} \sqrt{1 - \sqrt{x}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 i x^{2} \sqrt{1 - \sqrt{x}}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{8 i x^{2}}{- 3 x^{\frac{5}{2}} + 3 x^{2}} & \text{otherwise} \end{cases}}{4}
Simplificar
Gráfica
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Trazar el gráfico f(x)
Respuesta numérica [src] 
13.2910592040078
13.2910592040078

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

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