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Resolución de integrales/ (x+1)/ sqrt(x)/ sqrt(x)(x+1)^(-3)

Integral de sqrt(x)(x+1)^(-3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1             
 ----           
  5/6           
 a              
   /            
  |             
  |     ___     
  |   \/ x      
  |  -------- dx
  |         3   
  |  (x + 1)    
  |             
 /              
 5/6            
a               
----            
 4
∫a5641a56x(x+1)3 dx\int\limits_{\frac{a^{\frac{5}{6}}}{4}}^{\frac{1}{a^{\frac{5}{6}}}} \frac{\sqrt{x}}{\left(x + 1\right)^{3}}\, dx 
Integral(sqrt(x)/(x + 1)^3, (x, a^(5/6)/4, a^(-5/6)))
Respuesta (Indefinida) [src] 
  /                                             
 |                                              
 |    ___                /  ___\     ___        
 |  \/ x             atan\\/ x /   \/ x *(1 - x)
 | -------- dx = C + ----------- - -------------
 |        3               4                   2 
 | (x + 1)                           4*(1 + x)  
 |                                              
/
∫x(x+1)3 dx=C−x(1−x)4(x+1)2+atan⁡(x)4\int \frac{\sqrt{x}}{\left(x + 1\right)^{3}}\, dx = C - \frac{\sqrt{x} \left(1 - x\right)}{4 \left(x + 1\right)^{2}} + \frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{4}
Simplificar
Respuesta [src] 
                               /  1  \                                                              / 5/12\                                       /  1  \                   /  1  \                    / 5/12\                 / 5/12\   
                           atan|-----|                                                              |a    |                                   atan|-----|             2*atan|-----|            5/6     |a    |         5/3     |a    |   
                               | 5/12|              5/12                                        atan|-----|                5/4                    | 5/12|                   | 5/12|           a   *atan|-----|        a   *atan|-----|   
          1                    \a    /             a                          1                     \  2  /               a                       \a    /                   \a    /                    \  2  /                 \  2  /   
---------------------- + --------------- + --------------------- - ----------------------- - ----------------- - --------------------- + ---------------------- + ---------------------- - --------------------- - ----------------------
 5/4 /     4      8  \        4      8       /              5/3\    5/12 /     4      8  \                 5/3     /              5/3\    5/3 /     4      8  \    5/6 /     4      8  \     /              5/3\      /              5/3\
a   *|4 + ---- + ----|   4 + ---- + ----     |       5/6   a   |   a    *|4 + ---- + ----|          5/6   a        |       5/6   a   |   a   *|4 + ---- + ----|   a   *|4 + ---- + ----|     |       5/6   a   |      |       5/6   a   |
     |     5/3    5/6|        5/3    5/6   2*|4 + 2*a    + ----|         |     5/3    5/6|   4 + 2*a    + ----   8*|4 + 2*a    + ----|        |     5/3    5/6|        |     5/3    5/6|   2*|4 + 2*a    + ----|   16*|4 + 2*a    + ----|
     \    a      a   /       a      a        \              4  /         \    a      a   /                 4       \              4  /        \    a      a   /        \    a      a   /     \              4  /      \              4  /
a5122(2a56+a534+4)−a56atan⁡(a5122)2(2a56+a534+4)−a548(2a56+a534+4)−a53atan⁡(a5122)16(2a56+a534+4)−atan⁡(a5122)2a56+a534+4+atan⁡(1a512)4+4a53+8a56+atan⁡(1a512)a53(4+4a53+8a56)+1a54(4+4a53+8a56)+2atan⁡(1a512)a56(4+4a53+8a56)−1a512(4+4a53+8a56)\frac{a^{\frac{5}{12}}}{2 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{6}} \operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{2 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{4}}}{8 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{3}} \operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{16 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{\operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4} + \frac{\operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}} + \frac{\operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{a^{\frac{5}{3}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} + \frac{1}{a^{\frac{5}{4}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} + \frac{2 \operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{a^{\frac{5}{6}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} - \frac{1}{a^{\frac{5}{12}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} 
=
= 
                               /  1  \                                                              / 5/12\                                       /  1  \                   /  1  \                    / 5/12\                 / 5/12\   
                           atan|-----|                                                              |a    |                                   atan|-----|             2*atan|-----|            5/6     |a    |         5/3     |a    |   
                               | 5/12|              5/12                                        atan|-----|                5/4                    | 5/12|                   | 5/12|           a   *atan|-----|        a   *atan|-----|   
          1                    \a    /             a                          1                     \  2  /               a                       \a    /                   \a    /                    \  2  /                 \  2  /   
---------------------- + --------------- + --------------------- - ----------------------- - ----------------- - --------------------- + ---------------------- + ---------------------- - --------------------- - ----------------------
 5/4 /     4      8  \        4      8       /              5/3\    5/12 /     4      8  \                 5/3     /              5/3\    5/3 /     4      8  \    5/6 /     4      8  \     /              5/3\      /              5/3\
a   *|4 + ---- + ----|   4 + ---- + ----     |       5/6   a   |   a    *|4 + ---- + ----|          5/6   a        |       5/6   a   |   a   *|4 + ---- + ----|   a   *|4 + ---- + ----|     |       5/6   a   |      |       5/6   a   |
     |     5/3    5/6|        5/3    5/6   2*|4 + 2*a    + ----|         |     5/3    5/6|   4 + 2*a    + ----   8*|4 + 2*a    + ----|        |     5/3    5/6|        |     5/3    5/6|   2*|4 + 2*a    + ----|   16*|4 + 2*a    + ----|
     \    a      a   /       a      a        \              4  /         \    a      a   /                 4       \              4  /        \    a      a   /        \    a      a   /     \              4  /      \              4  /
a5122(2a56+a534+4)−a56atan⁡(a5122)2(2a56+a534+4)−a548(2a56+a534+4)−a53atan⁡(a5122)16(2a56+a534+4)−atan⁡(a5122)2a56+a534+4+atan⁡(1a512)4+4a53+8a56+atan⁡(1a512)a53(4+4a53+8a56)+1a54(4+4a53+8a56)+2atan⁡(1a512)a56(4+4a53+8a56)−1a512(4+4a53+8a56)\frac{a^{\frac{5}{12}}}{2 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{6}} \operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{2 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{4}}}{8 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{3}} \operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{16 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{\operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4} + \frac{\operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}} + \frac{\operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{a^{\frac{5}{3}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} + \frac{1}{a^{\frac{5}{4}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} + \frac{2 \operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{a^{\frac{5}{6}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} - \frac{1}{a^{\frac{5}{12}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} 
1/(a^(5/4)*(4 + 4/a^(5/3) + 8/a^(5/6))) + atan(a^(-5/12))/(4 + 4/a^(5/3) + 8/a^(5/6)) + a^(5/12)/(2*(4 + 2*a^(5/6) + a^(5/3)/4)) - 1/(a^(5/12)*(4 + 4/a^(5/3) + 8/a^(5/6))) - atan(a^(5/12)/2)/(4 + 2*a^(5/6) + a^(5/3)/4) - a^(5/4)/(8*(4 + 2*a^(5/6) + a^(5/3)/4)) + atan(a^(-5/12))/(a^(5/3)*(4 + 4/a^(5/3) + 8/a^(5/6))) + 2*atan(a^(-5/12))/(a^(5/6)*(4 + 4/a^(5/3) + 8/a^(5/6))) - a^(5/6)*atan(a^(5/12)/2)/(2*(4 + 2*a^(5/6) + a^(5/3)/4)) - a^(5/3)*atan(a^(5/12)/2)/(16*(4 + 2*a^(5/6) + a^(5/3)/4))

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

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