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Resolución de integrales/ ln^2(x)/ sqrt(x)/ sqrt(x)ln^2(x)

Integral de sqrt(x)ln^2(x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  o                 
  /                 
 |                  
 |    ___    2      
 |  \/ x *log (x) dx
 |                  
/                   
0
∫0oxlog⁡(x)2 dx\int\limits_{0}^{o} \sqrt{x} \log{\left(x \right)}^{2}\, dx 
Integral(sqrt(x)*log(x)^2, (x, 0, o))
Respuesta (Indefinida) [src] 
  /                                                               
 |                            3/2      3/2             3/2    2   
 |   ___    2             16*x      8*x   *log(x)   2*x   *log (x)
 | \/ x *log (x) dx = C + ------- - ------------- + --------------
 |                           27           9               3       
/
∫xlog⁡(x)2 dx=C+2x32log⁡(x)23−8x32log⁡(x)9+16x3227\int \sqrt{x} \log{\left(x \right)}^{2}\, dx = C + \frac{2 x^{\frac{3}{2}} \log{\left(x \right)}^{2}}{3} - \frac{8 x^{\frac{3}{2}} \log{\left(x \right)}}{9} + \frac{16 x^{\frac{3}{2}}}{27}
Simplificar
Respuesta [src] 
/                             3/2    /    1     \                                                 3/2    2/    1     \                                              3/2    /1\           3/2    2/1\           
|           3/2   8*Max(-1, o)   *log|----------|               3/2                   2*Max(-1, o)   *log |----------|               3/2    2               8*I*(-o)   *log|-|   2*I*(-o)   *log |-|           
|  16*I*(-o)                         \Max(-1, o)/   8*Max(-1, o)   *log(Max(-1, o))                       \Max(-1, o)/   2*Max(-1, o)   *log (Max(-1, o))                  \o/                   \o/           
<- ------------ - ------------------------------- - ------------------------------- - -------------------------------- + -------------------------------- - ------------------ - -------------------  for o < 0
|       27                       9                                 9                                 3                                  3                           9                     3                    
|                                                                                                                                                                                                              
\                                                                                                nan                                                                                                  otherwise
{−2i(−o)32log⁡(1o)23−8i(−o)32log⁡(1o)9−16i(−o)3227−2log⁡(1max⁡(−1,o))2max⁡(−1,o)323−8log⁡(1max⁡(−1,o))max⁡(−1,o)329+2log⁡(max⁡(−1,o))2max⁡(−1,o)323−8log⁡(max⁡(−1,o))max⁡(−1,o)329for o<0NaNotherwise\begin{cases} - \frac{2 i \left(- o\right)^{\frac{3}{2}} \log{\left(\frac{1}{o} \right)}^{2}}{3} - \frac{8 i \left(- o\right)^{\frac{3}{2}} \log{\left(\frac{1}{o} \right)}}{9} - \frac{16 i \left(- o\right)^{\frac{3}{2}}}{27} - \frac{2 \log{\left(\frac{1}{\max\left(-1, o\right)} \right)}^{2} \max\left(-1, o\right)^{\frac{3}{2}}}{3} - \frac{8 \log{\left(\frac{1}{\max\left(-1, o\right)} \right)} \max\left(-1, o\right)^{\frac{3}{2}}}{9} + \frac{2 \log{\left(\max\left(-1, o\right) \right)}^{2} \max\left(-1, o\right)^{\frac{3}{2}}}{3} - \frac{8 \log{\left(\max\left(-1, o\right) \right)} \max\left(-1, o\right)^{\frac{3}{2}}}{9} & \text{for}\: o < 0 \\\text{NaN} & \text{otherwise} \end{cases} 
=
= 
/                             3/2    /    1     \                                                 3/2    2/    1     \                                              3/2    /1\           3/2    2/1\           
|           3/2   8*Max(-1, o)   *log|----------|               3/2                   2*Max(-1, o)   *log |----------|               3/2    2               8*I*(-o)   *log|-|   2*I*(-o)   *log |-|           
|  16*I*(-o)                         \Max(-1, o)/   8*Max(-1, o)   *log(Max(-1, o))                       \Max(-1, o)/   2*Max(-1, o)   *log (Max(-1, o))                  \o/                   \o/           
<- ------------ - ------------------------------- - ------------------------------- - -------------------------------- + -------------------------------- - ------------------ - -------------------  for o < 0
|       27                       9                                 9                                 3                                  3                           9                     3                    
|                                                                                                                                                                                                              
\                                                                                                nan                                                                                                  otherwise
{−2i(−o)32log⁡(1o)23−8i(−o)32log⁡(1o)9−16i(−o)3227−2log⁡(1max⁡(−1,o))2max⁡(−1,o)323−8log⁡(1max⁡(−1,o))max⁡(−1,o)329+2log⁡(max⁡(−1,o))2max⁡(−1,o)323−8log⁡(max⁡(−1,o))max⁡(−1,o)329for o<0NaNotherwise\begin{cases} - \frac{2 i \left(- o\right)^{\frac{3}{2}} \log{\left(\frac{1}{o} \right)}^{2}}{3} - \frac{8 i \left(- o\right)^{\frac{3}{2}} \log{\left(\frac{1}{o} \right)}}{9} - \frac{16 i \left(- o\right)^{\frac{3}{2}}}{27} - \frac{2 \log{\left(\frac{1}{\max\left(-1, o\right)} \right)}^{2} \max\left(-1, o\right)^{\frac{3}{2}}}{3} - \frac{8 \log{\left(\frac{1}{\max\left(-1, o\right)} \right)} \max\left(-1, o\right)^{\frac{3}{2}}}{9} + \frac{2 \log{\left(\max\left(-1, o\right) \right)}^{2} \max\left(-1, o\right)^{\frac{3}{2}}}{3} - \frac{8 \log{\left(\max\left(-1, o\right) \right)} \max\left(-1, o\right)^{\frac{3}{2}}}{9} & \text{for}\: o < 0 \\\text{NaN} & \text{otherwise} \end{cases} 
Piecewise((-16*i*(-o)^(3/2)/27 - 8*Max(-1, o)^(3/2)*log(1/Max(-1, o))/9 - 8*Max(-1, o)^(3/2)*log(Max(-1, o))/9 - 2*Max(-1, o)^(3/2)*log(1/Max(-1, o))^2/3 + 2*Max(-1, o)^(3/2)*log(Max(-1, o))^2/3 - 8*i*(-o)^(3/2)*log(1/o)/9 - 2*i*(-o)^(3/2)*log(1/o)^2/3, o < 0), (nan, True))

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

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