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Resolución de integrales/ y^(3/2)/ 1/(y^(3/2)+c)

Integral de 1/(y^(3/2)+c) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |     1       
 |  -------- dy
 |   3/2       
 |  y    + c   
 |             
/              
0
∫011c+y32 dy\int\limits_{0}^{1} \frac{1}{c + y^{\frac{3}{2}}}\, dy 
Integral(1/(y^(3/2) + c), (y, 0, 1))
Respuesta (Indefinida) [src] 
                     //                                                                                                                       /  ___         2/3   ___   ___\            \
                     ||                                                                                                         2/3   ___     |\/ 3    2*(-1)   *\/ 3 *\/ y |            |
                     ||                                                                                                   2*(-1)   *\/ 3 *atan|----- - ---------------------|            |
  /                  ||        2/3    /  ___   3 ____ 3 ___\       2/3    /            2/3  2/3     3 ____ 3 ___   ___\                       |  3              3 ___       |            |
 |                   ||  2*(-1)   *log\\/ y  - \/ -1 *\/ c /   (-1)   *log\4*y + 4*(-1)   *c    + 4*\/ -1 *\/ c *\/ y /                       \               3*\/ c        /            |
 |    1              ||- ----------------------------------- + -------------------------------------------------------- - ---------------------------------------------------  for c != 0|
 | -------- dy = C + |<                  3 ___                                           3 ___                                                    3 ___                                  |
 |  3/2              ||                3*\/ c                                          3*\/ c                                                   3*\/ c                                   |
 | y    + c          ||                                                                                                                                                                  |
 |                   ||                                                                         -2                                                                                       |
/                    ||                                                                        -----                                                                           otherwise |
                     ||                                                                          ___                                                                                     |
                     \\                                                                        \/ y                                                                                      /
∫1c+y32 dy=C+{−2(−1)23log⁡(−−13c3+y)3c3+(−1)23log⁡(4(−1)23c23+4−13c3y+4y)3c3−2(−1)233atan⁡(33−2(−1)233y3c3)3c3for c≠0−2yotherwise\int \frac{1}{c + y^{\frac{3}{2}}}\, dy = C + \begin{cases} - \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(- \sqrt[3]{-1} \sqrt[3]{c} + \sqrt{y} \right)}}{3 \sqrt[3]{c}} + \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \left(-1\right)^{\frac{2}{3}} c^{\frac{2}{3}} + 4 \sqrt[3]{-1} \sqrt[3]{c} \sqrt{y} + 4 y \right)}}{3 \sqrt[3]{c}} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3} \sqrt{y}}{3 \sqrt[3]{c}} \right)}}{3 \sqrt[3]{c}} & \text{for}\: c \neq 0 \\- \frac{2}{\sqrt{y}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                                                                                                                                                                        /  ___         2/3   ___\                                                     
|                                                                                                                                                          2/3   ___     |\/ 3    2*(-1)   *\/ 3 |                                                     
|                                                                                                                                                    2*(-1)   *\/ 3 *atan|----- - ---------------|                                                     
|        2/3    /    3 ____ 3 ___\       2/3    /      2/3  2/3\       2/3    /      3 ____ 3 ___         2/3  2/3\         2/3    / 3 ____ 3 ___\                       |  3           3 ___    |          2/3   ___                                  
|  2*(-1)   *log\1 - \/ -1 *\/ c /   (-1)   *log\4*(-1)   *c   /   (-1)   *log\4 + 4*\/ -1 *\/ c  + 4*(-1)   *c   /   2*(-1)   *log\-\/ -1 *\/ c /                       \            3*\/ c     /   pi*(-1)   *\/ 3                                   
<- ------------------------------- - --------------------------- + ------------------------------------------------ + ---------------------------- - --------------------------------------------- + ----------------  for And(c > -oo, c < oo, c != 0)
|                3 ___                           3 ___                                   3 ___                                    3 ___                                   3 ___                            3 ___                                       
|              3*\/ c                          3*\/ c                                  3*\/ c                                   3*\/ c                                  3*\/ c                           9*\/ c                                        
|                                                                                                                                                                                                                                                      
|                                                                                                         oo                                                                                                                      otherwise            
\
{2(−1)23log⁡(−−13c3)3c3−(−1)23log⁡(4(−1)23c23)3c3−2(−1)23log⁡(−−13c3+1)3c3+(−1)23log⁡(4(−1)23c23+4−13c3+4)3c3−2(−1)233atan⁡(33−2(−1)2333c3)3c3+(−1)233π9c3for c>−∞∧c<∞∧c≠0∞otherwise\begin{cases} \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(- \sqrt[3]{-1} \sqrt[3]{c} \right)}}{3 \sqrt[3]{c}} - \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \left(-1\right)^{\frac{2}{3}} c^{\frac{2}{3}} \right)}}{3 \sqrt[3]{c}} - \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(- \sqrt[3]{-1} \sqrt[3]{c} + 1 \right)}}{3 \sqrt[3]{c}} + \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \left(-1\right)^{\frac{2}{3}} c^{\frac{2}{3}} + 4 \sqrt[3]{-1} \sqrt[3]{c} + 4 \right)}}{3 \sqrt[3]{c}} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3}}{3 \sqrt[3]{c}} \right)}}{3 \sqrt[3]{c}} + \frac{\left(-1\right)^{\frac{2}{3}} \sqrt{3} \pi}{9 \sqrt[3]{c}} & \text{for}\: c > -\infty \wedge c < \infty \wedge c \neq 0 \\\infty & \text{otherwise} \end{cases} 
=
= 
/                                                                                                                                                                        /  ___         2/3   ___\                                                     
|                                                                                                                                                          2/3   ___     |\/ 3    2*(-1)   *\/ 3 |                                                     
|                                                                                                                                                    2*(-1)   *\/ 3 *atan|----- - ---------------|                                                     
|        2/3    /    3 ____ 3 ___\       2/3    /      2/3  2/3\       2/3    /      3 ____ 3 ___         2/3  2/3\         2/3    / 3 ____ 3 ___\                       |  3           3 ___    |          2/3   ___                                  
|  2*(-1)   *log\1 - \/ -1 *\/ c /   (-1)   *log\4*(-1)   *c   /   (-1)   *log\4 + 4*\/ -1 *\/ c  + 4*(-1)   *c   /   2*(-1)   *log\-\/ -1 *\/ c /                       \            3*\/ c     /   pi*(-1)   *\/ 3                                   
<- ------------------------------- - --------------------------- + ------------------------------------------------ + ---------------------------- - --------------------------------------------- + ----------------  for And(c > -oo, c < oo, c != 0)
|                3 ___                           3 ___                                   3 ___                                    3 ___                                   3 ___                            3 ___                                       
|              3*\/ c                          3*\/ c                                  3*\/ c                                   3*\/ c                                  3*\/ c                           9*\/ c                                        
|                                                                                                                                                                                                                                                      
|                                                                                                         oo                                                                                                                      otherwise            
\
{2(−1)23log⁡(−−13c3)3c3−(−1)23log⁡(4(−1)23c23)3c3−2(−1)23log⁡(−−13c3+1)3c3+(−1)23log⁡(4(−1)23c23+4−13c3+4)3c3−2(−1)233atan⁡(33−2(−1)2333c3)3c3+(−1)233π9c3for c>−∞∧c<∞∧c≠0∞otherwise\begin{cases} \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(- \sqrt[3]{-1} \sqrt[3]{c} \right)}}{3 \sqrt[3]{c}} - \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \left(-1\right)^{\frac{2}{3}} c^{\frac{2}{3}} \right)}}{3 \sqrt[3]{c}} - \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(- \sqrt[3]{-1} \sqrt[3]{c} + 1 \right)}}{3 \sqrt[3]{c}} + \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \left(-1\right)^{\frac{2}{3}} c^{\frac{2}{3}} + 4 \sqrt[3]{-1} \sqrt[3]{c} + 4 \right)}}{3 \sqrt[3]{c}} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3}}{3 \sqrt[3]{c}} \right)}}{3 \sqrt[3]{c}} + \frac{\left(-1\right)^{\frac{2}{3}} \sqrt{3} \pi}{9 \sqrt[3]{c}} & \text{for}\: c > -\infty \wedge c < \infty \wedge c \neq 0 \\\infty & \text{otherwise} \end{cases} 
Piecewise((-2*(-1)^(2/3)*log(1 - (-1)^(1/3)*c^(1/3))/(3*c^(1/3)) - (-1)^(2/3)*log(4*(-1)^(2/3)*c^(2/3))/(3*c^(1/3)) + (-1)^(2/3)*log(4 + 4*(-1)^(1/3)*c^(1/3) + 4*(-1)^(2/3)*c^(2/3))/(3*c^(1/3)) + 2*(-1)^(2/3)*log(-(-1)^(1/3)*c^(1/3))/(3*c^(1/3)) - 2*(-1)^(2/3)*sqrt(3)*atan(sqrt(3)/3 - 2*(-1)^(2/3)*sqrt(3)/(3*c^(1/3)))/(3*c^(1/3)) + pi*(-1)^(2/3)*sqrt(3)/(9*c^(1/3)), (c > -oo)∧(c < oo)∧(Ne(c, 0))), (oo, True))

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

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