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Resolución de integrales/ x^2*sin(p*x/3)

Integral de x^2*sin(p*x/3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  3               
  /               
 |                
 |   2    /p*x\   
 |  x *sin|---| dx
 |        \ 3 /   
 |                
/                 
0
$$\int\limits_{0}^{3} x^{2} \sin{\left(\frac{p x}{3} \right)}\, dx$$ 
Integral(x^2*sin((p*x)/3), (x, 0, 3))
Respuesta (Indefinida) [src] 
                          //                     0                       for p = 0\                               
                          ||                                                      |                               
                          ||   //     /p*x\          /p*x\            \           |                               
                          ||   ||9*cos|---|   3*x*sin|---|            |           |                               
  /                       ||   ||     \ 3 /          \ 3 /            |           |      //     0       for p = 0\
 |                        ||   ||---------- + ------------  for p != 0|           |      ||                      |
 |  2    /p*x\            ||   ||     2            p                  |           |    2 ||      /p*x\           |
 | x *sin|---| dx = C - 2*|<-3*|<    p                                |           | + x *|<-3*cos|---|           |
 |       \ 3 /            ||   ||                                     |           |      ||      \ 3 /           |
 |                        ||   ||            2                        |           |      ||-----------  otherwise|
/                         ||   ||           x                         |           |      \\     p                /
                          ||   ||           --              otherwise |           |                               
                          ||   \\           2                         /           |                               
                          ||-------------------------------------------  otherwise|                               
                          \\                     p                                /
$$\int x^{2} \sin{\left(\frac{p x}{3} \right)}\, dx = C + x^{2} \left(\begin{cases} 0 & \text{for}\: p = 0 \\- \frac{3 \cos{\left(\frac{p x}{3} \right)}}{p} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} 0 & \text{for}\: p = 0 \\- \frac{3 \left(\begin{cases} \frac{3 x \sin{\left(\frac{p x}{3} \right)}}{p} + \frac{9 \cos{\left(\frac{p x}{3} \right)}}{p^{2}} & \text{for}\: p \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}\right)}{p} & \text{otherwise} \end{cases}\right)$$
Simplificar
Respuesta [src] 
/  54   27*cos(p)   54*cos(p)   54*sin(p)                                  
|- -- - --------- + --------- + ---------  for And(p > -oo, p < oo, p != 0)
|   3       p            3           2                                     
<  p                    p           p                                      
|                                                                          
|                   0                                 otherwise            
\
$$\begin{cases} - \frac{27 \cos{\left(p \right)}}{p} + \frac{54 \sin{\left(p \right)}}{p^{2}} + \frac{54 \cos{\left(p \right)}}{p^{3}} - \frac{54}{p^{3}} & \text{for}\: p > -\infty \wedge p < \infty \wedge p \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
=
= 
/  54   27*cos(p)   54*cos(p)   54*sin(p)                                  
|- -- - --------- + --------- + ---------  for And(p > -oo, p < oo, p != 0)
|   3       p            3           2                                     
<  p                    p           p                                      
|                                                                          
|                   0                                 otherwise            
\
$$\begin{cases} - \frac{27 \cos{\left(p \right)}}{p} + \frac{54 \sin{\left(p \right)}}{p^{2}} + \frac{54 \cos{\left(p \right)}}{p^{3}} - \frac{54}{p^{3}} & \text{for}\: p > -\infty \wedge p < \infty \wedge p \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
Piecewise((-54/p^3 - 27*cos(p)/p + 54*cos(p)/p^3 + 54*sin(p)/p^2, (p > -oo)∧(p < oo)∧(Ne(p, 0))), (0, True))

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

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