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Resolución de integrales/ x-0,5/ 0,5*x/ (x-0,5*x*x)*sin(pi*m*x/2)

Integral de (x-0,5*x*x)*sin(pi*m*x/2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  2                         
  /                         
 |                          
 |  /    x  \    /pi*m*x\   
 |  |x - -*x|*sin|------| dx
 |  \    2  /    \  2   /   
 |                          
/                           
0
∫02(−x2x+x)sin⁡(xπm2) dx\int\limits_{0}^{2} \left(- \frac{x}{2} x + x\right) \sin{\left(\frac{x \pi m}{2} \right)}\, dx 
Integral((x - x/2*x)*sin(((pi*m)*x)/2), (x, 0, 2))
Respuesta (Indefinida) [src] 
                                                                                                                         //      0         for m = 0\   //                        0                          for m = 0\
                                                                                                                         ||                         |   ||                                                            |
                                  //                0                   for m = 0\                                     2 ||      /pi*m*x\           |   ||   //     /pi*m*x\          /pi*m*x\            \           |
                                  ||                                             |                                    x *|<-2*cos|------|           |   ||   ||4*cos|------|   2*x*sin|------|            |           |
  /                               ||   //     /pi*m*x\               \           |     //      0         for m = 0\      ||      \  2   /           |   ||   ||     \  2   /          \  2   /            |           |
 |                                ||   ||2*sin|------|               |           |     ||                         |      ||--------------  otherwise|   ||   ||------------- + ---------------  for m != 0|           |
 | /    x  \    /pi*m*x\          ||   ||     \  2   /      pi*m     |           |     ||      /pi*m*x\           |      \\     pi*m                /   ||   ||      2  2            pi*m                 |           |
 | |x - -*x|*sin|------| dx = C - |<-2*|<-------------  for ---- != 0|           | + x*|<-2*cos|------|           | - ------------------------------- + |<-2*|<    pi *m                                  |           |
 | \    2  /    \  2   /          ||   ||     pi*m           2       |           |     ||      \  2   /           |                  2                  ||   ||                                           |           |
 |                                ||   ||                            |           |     ||--------------  otherwise|                                     ||   ||               2                           |           |
/                                 ||   \\      x          otherwise  /           |     \\     pi*m                /                                     ||   ||              x                            |           |
                                  ||----------------------------------  otherwise|                                                                      ||   ||              --                 otherwise |           |
                                  \\               pi*m                          /                                                                      ||   \\              2                            /           |
                                                                                                                                                        ||-------------------------------------------------  otherwise|
                                                                                                                                                        \\                       pi*m                                 /
∫(−x2x+x)sin⁡(xπm2) dx=C−x2({0for m=0−2cos⁡(πmx2)πmotherwise)2+x({0for m=0−2cos⁡(πmx2)πmotherwise)−{0for m=0−2({2sin⁡(πmx2)πmfor πm2≠0xotherwise)πmotherwise+{0for m=0−2({2xsin⁡(πmx2)πm+4cos⁡(πmx2)π2m2for m≠0x22otherwise)πmotherwise\int \left(- \frac{x}{2} x + x\right) \sin{\left(\frac{x \pi m}{2} \right)}\, dx = C - \frac{x^{2} \left(\begin{cases} 0 & \text{for}\: m = 0 \\- \frac{2 \cos{\left(\frac{\pi m x}{2} \right)}}{\pi m} & \text{otherwise} \end{cases}\right)}{2} + x \left(\begin{cases} 0 & \text{for}\: m = 0 \\- \frac{2 \cos{\left(\frac{\pi m x}{2} \right)}}{\pi m} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: m = 0 \\- \frac{2 \left(\begin{cases} \frac{2 \sin{\left(\frac{\pi m x}{2} \right)}}{\pi m} & \text{for}\: \frac{\pi m}{2} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{\pi m} & \text{otherwise} \end{cases} + \begin{cases} 0 & \text{for}\: m = 0 \\- \frac{2 \left(\begin{cases} \frac{2 x \sin{\left(\frac{\pi m x}{2} \right)}}{\pi m} + \frac{4 \cos{\left(\frac{\pi m x}{2} \right)}}{\pi^{2} m^{2}} & \text{for}\: m \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}\right)}{\pi m} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/  8      8*cos(pi*m)   4*sin(pi*m)                                  
|------ - ----------- - -----------  for And(m > -oo, m < oo, m != 0)
|  3  3        3  3          2  2
{−4sin⁡(πm)π2m2−8cos⁡(πm)π3m3+8π3m3for m>−∞∧m<∞∧m≠00otherwise\begin{cases} - \frac{4 \sin{\left(\pi m \right)}}{\pi^{2} m^{2}} - \frac{8 \cos{\left(\pi m \right)}}{\pi^{3} m^{3}} + \frac{8}{\pi^{3} m^{3}} & \text{for}\: m > -\infty \wedge m < \infty \wedge m \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/  8      8*cos(pi*m)   4*sin(pi*m)                                  
|------ - ----------- - -----------  for And(m > -oo, m < oo, m != 0)
|  3  3        3  3          2  2
{−4sin⁡(πm)π2m2−8cos⁡(πm)π3m3+8π3m3for m>−∞∧m<∞∧m≠00otherwise\begin{cases} - \frac{4 \sin{\left(\pi m \right)}}{\pi^{2} m^{2}} - \frac{8 \cos{\left(\pi m \right)}}{\pi^{3} m^{3}} + \frac{8}{\pi^{3} m^{3}} & \text{for}\: m > -\infty \wedge m < \infty \wedge m \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((8/(pi^3*m^3) - 8*cos(pi*m)/(pi^3*m^3) - 4*sin(pi*m)/(pi^2*m^2), (m > -oo)∧(m < oo)∧(Ne(m, 0))), (0, True))

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

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